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The Liar Paradox (and other beasties)
Reference and Self-Reference
The philosophical theme of reference is no doubt a wide and deep ocean. My attempts at presenting a new perspective on the subject can certainly not be considered as the final word on the subject. [see my thread Truth and Necessity]. Reading Quine "The Ways of Paradox" (1966), I realized that my (Strawsonian) conception that language does not refer poses special problems when the objects of reference are themselves linguistic elements. As Juergen Habermas would say, (natural) language is its own metalanguage, and just like our mind, seems to be able to look upon itself.
Self-reference is not only the source of many antinomies, it could mean the negation of my analysis as a whole: if language does not refer, how could it ever self-refer? My aim is quite simple. I will try to show that self-reference is not possible. The line of argumentation is easy to follow: no self-reference without reference. And since reference is not a linguistic property but an action undertaken by the people concerned, action which can include, or solely consist of the use of appropriate linguistic tools, linguistic self-reference is a philosophical delusion.
I found the starting example in "Liar Paradox" (by Beal&Glanzberg in: Stanford Encyclopedia of Philosophy) a very appropriate one.

A) The first sentence in this essay is a lie.

Can a sentence refer to itself? If it can, why couldn't it refer to other objects?
Let us say my analysis that no sentence can ever speak meaningfully of the existence of the objects it refers to is a correct one. Does it also hold when the object of reference is itself a sentence, the same or a different one?
What does 'The first sentence in this essay is a lie" refer to? And can we doubt of the existence of this sentence?

We must distinguish between the object whose existence we can verify empirically, and the sentence we utter. Drawing on the Strawsonian distinction of meaning and use we could say that we, as observers, have two objects of study. 
1) The empirical sentence which is the sequence of linguistic signs on paper or on a screen.
2) The sentence [which happens to be made of the same signs as the first one], which can be uttered in numerous and distinct occasions.

When considering (1), the empirical object, we ask ourselves what the authors might have meant with this sentence. Obviously, they do not expect us to believe it but intend the sentence as an illustration of the Liar Paradox, as the title of their article states. We are allowed to hold our belief as to the validity of the assertion and consider it without passion. To be clear, we are not uttering it but putting it under an analytical microscope. As such it does not create any paradox.

Let us now consider it as an utterance, that is as some form of assertion which goes beyond merely reading it aloud: 
As such it represents something the speaker want us to understand and relate to. We still have two objects of analysis:

A)  The first sentence in this essay is a lie. [an empirical object]
B) 'The first sentence in this essay is a lie'. [an utterance]

[Speaker=s, Hearer=h; quotes, single and double, have been deliberately omitted as befits a natural dialogue.] 

s: The first sentence in this essay is a lie.
h: which essay?
s: this one.
h: it says The first sentence in this essay is a lie.
s: that is what I said.
h: no, you said, the first sentence in this essay is a lie, is a lie.
s: okay, whatever! prove me wrong!
h: what made you say that?
s: uh, to prove a paradox?
h: which paradox?
s: Oh, come on! What about I tell you that the sentence I am uttering right now is false?
h: which sentence?
s: the one I am uttering right now!
h: I am still waiting for you to utter it.
s: I just did!
h: when?
s: when I said the sentence I am uttering right now is false!
h: are you speaking of what you said when you said the sentence I am uttering right now is false, or of the sentence the sentence I am uttering right now is false?
s: uh, both?
h: okay, lets us start with the first part. What did you mean when you said the sentence I am uttering right now is false?
s: One last time, I was trying to show that language leads to contradictions, paradoxes, antinomies, or whatever you want to call them!
h: no need to shout, I get it, I really do. So you wanted to draw attention to a property of language, whatever it is.
s: yes! But not whatever property...
h: We will come back to that, I promise. let us look now at the sentence this sentence I am uttering right now is false. What do you think is meant by this and right now?
s: Oh my God! I can't believe it!

[If you want to react, please avoid complex formulas. They tend to make me feel stupid, so I just skip them.]

The Liar Paradox (and other beasties)
Reference and Existence

1) the set of all sets that are not members of themselves
only poses a problem if 'the set' refers to an object , or whether, as a linguistic element, it functions as a name or description, whether it  is simply used to refer to 'all sets that are not members of themselves". 

Let us concede that "all sets that are not members of themselves" form a multitude of real existing objects. The question now is whether "the set" is also an existing object, or whether it is simply the name or description of these real existing objects.

When we name somebody 'John', we do not create a new entity distinct of John. We cannot say that there is in the world John, and also 'John'.
What is peculiar is that the passage from name/description can be made surreptitiously. That is quite evident in:

2) The set of all sets that are not members of themselves exists.

The ambiguity disappears completely in:

3) The set of all sets that are not members of themselves is a misnomer.

The Liar Paradox (and other beasties)
It is indeed important to determine whether any paradox is a genuine paradox before one attempts to resolve the paradox. The liar's paradox has withstood the test of time and seems to be a genuine paradox. So, a new attempt to demonstrate that it is not a genuine paradox must be looked at keenly.
You begin with Beall and Glanzberg's Stanford Encyclopedia entry on Liar's paradox. You must remember that Stanford Encyclopedia articles are written for a wide audience. Hence, they begin with a catchy 'The first sentence in this essay is a lie'. This will hold a reader's attention at the outset but this is clearly not the type of sentence that has been used to generate the liar's paradox. 'lie' involves an intentional context. The truth-value of such a sentence is not easy to determine. 

The more perspicuous sentence is stated in sections 1.1:

FLiar:FLiar is false.

And then in 1.2:

ULiar is not true.

The liar paradox is now generated. 

Tarski puts it this way:

The sentence printed in this paper on p. 339, l. 11, is not true.
For brevity we shall replace the sentence just stated by the letter `s.'

According to our convention concerning the adequate usage of the
term ``true,'' we assert the following equivalence of the form (T):
(1) `s' is true if, and only if, the sentence printed in this paper on
p. 339, l. 11, is not true.  (Lynch, The Nature of Truth, 2001, p. 339).

From this the liar paradox is generated. I have summarized Tarski's section 7 of his article in this way:

1.      The Antinomy of the Liar

Ø  the sentence printed in this document on p. 5, l. 28, is not true.

Ø  Let s be short for ‘the sentence printed in this document on p. 5, l. 28, is not true.

Ø  According to equivalence T:

(1)   ‘s’ is true iff the sentence printed in this document on p. 5, l. 28, is not true.

(2)   ‘s is identical with the sentence printed in this document on p. 5, l. 28.

(3)   ‘s’ is true iff ‘s’ is not true.

Ø  (3) is derived from (1) and (2) by using substitution of identicals.

Ø  (3) is an obvious contradiction.

Ø  Antinomies are very important in the development of formal systems.

Ø  The antinomy of the liar is important for the development of formal semantics.

Ø  But first, the antinomy has to be stated formally.

Ø  Second, we must resolve the antinomy.

ü  We must examine the premises on which the antinomy is based.

ü  We must then reject one of these premises.

ü  We must then look at the consequences of rejecting this premise.


I may have made some errors in my presentation of Tarski. But the formal statement of the antinomy is crucial. 

Unfortunately, your dialogue is based on the first sentence of the Stanford entry not on the first formally stated sentence of the antinomy.

So, I do not think you have successfully demonstrated that there is no self-referential paradox being generated here.

Finally, I just don't see why you need to bring in Habermas. Many have said this about natural language, it is hardly a profound statement. If Habermas has worked with the formal generation of the paradox and has demonstrated that this cannot be done, I would indeed be curious. Perhaps you can enlighten us on where he has done so. Otherwise, your reference to him is neither here nor there as he is neither a worker in philosophy of logic nor in theories of truth.


Priyedarshi Jetli

The Liar Paradox (and other beasties)
Barber Paradox

[A short fictive dialogue between a detective (d) and a suspect known as Jack (j).]

d: your whiskers are well tended, and your lips and kin are exceptionally smooth. May I ask who your barber is?
j: I'm afraid I'm my own barber.
d: are you really?
j: I confess that I am somewhat paranoid. I do not trust anybody with a knife in the vicinity of my throat.
d: you must be very good with knives, then.
j: I can shave myself, if that's what you mean.

Could Jack be the barber that everyone is talking about? The one that shaves all men in the village that do not shave themselves? I see no reason why not. Two reasons for that:

1) you can be your own barber,
2) you do not have to be a barber to shave yourself.

The Barber paradox only makes sense if we make another assumption:
3) If you shave yourself, you are not a barber.

When the barber shaves himself, he does what men throughout history have done: shave themselves. The paradox rests on the false assumption that when he is shaving himself he is still behaving as a barber.
Being a barber is the job or function of the person we are referring to. But in the apparent paradox, the term gains an existence of its own and replaces the real person as it were. 'Barber' becomes its own reference.

Last but not least, 'the barber' in this apparent paradox behaves like a logical variable, and logicians are very attached to the rule that states that one may not change the meaning of a variable during an argumentation. A rule which of course is happily violated anytime a real discussion in natural language takes place.
Quine's solution, that such a barber as depicted by the classic paradox cannot exist, is the result of a very poor analysis of natural language rhetorics. He takes the formulation of the paradox at its literal value and sees then no other way to save Logic (as a human faculty and as a discipline) than to magically declare the barber as a non-entity.

To sum up:
"The barber" is not the person who does the shaving, but the name or description we give of him. Kripke made abundantly clear that a description is other than a name, and that a person, miraculously, could turn out to be someone entirely in some alternate time line. Whatever conception strikes your fancy, "the barber" remains a string of sounds to which we first must give a meaning, and not necessarily the same meaning each time.

The Liar Paradox (and other beasties)
Grelling's Paradox

1) is 'heterological' heterological?
2) is 'true of self' true of self?
3) 'short' is short
4) 'polysyllabic' is polysyllabic

According to Quine the whole discussion revolves around the matter of the truth of predicates: "This principle simply reflects what we mean in saying that adjectives are true of things. It is a hard principle to distrust, and yet it is obviously the principle that is to blame for our antinomy."

This quote sums up really nicely Quine's blind spot.
To show that I need a very simple sentence,

5) is 'red' red?

If you heard someone uttering (5), your first reaction would certainly be:

6) is a red what red?

In other words, 'red' in (5) only makes sense when applied to an object. Following this principle, to what object respectively are the sentences 1-4 applied to?

Let us take

3) 'short' is short.

(3) is obviously true only if you understand it as being applied to the word and not to the meaning 'short' since that would yield the following absurdities:

30) the meaning of short is short.

That does not sound too bad, does it?
how about:

31) the meaning of short is five letters long.

We would immediately recognize that what we mean is in fact:

32) 'short' is five letters long.

Obviously when dealing with (1) and (2), we are making the same mistake as with 'red' or short. We are confusing each time the word, an empirical object that can be written, uttered or just read, and the meaning we give to this object.

In both sentences: 
1) is 'heterological' heterological?
2) is 'true of self' true of self?

we are asked to carry judgment on an ambiguous event. 'heterological' and 'true of self' are just like 'red', you can apply them to other objects, but not to themselves.

Before you protest let me be a little more precise:
Both (1) and (2) are linguistically correct. So it would seem that we can ask of 'heterological' if it is heterological. But what we should be saying is

10) is the meaning of 'heterological' heterological?
20) is the meaning of 'true of self' true of self?

Both these last sentences have, as far as I can see, as much meaning as:

60) is the meaning of 'red' red?

We can of course always choose for the other interpretation:

11) is the word 'heterological' heterological?
21) is the word 'true of self' true of self?
61) is the word 'red' red?

The fact that

33) is the word 'short' short?

can be answered meaningfully, does not inoculate the previous sentences against non-sense.

The Liar Paradox (and other beasties)
Separation Angst

"Separation: For every set A and every given property, there is a set containing exactly the elements of A that have that property. A property is given by a formula f of the first-order language of set theory." (Stanford Encyclopedia of Philosophy)

One property could be: 'is not a member of itself".

We would then have a set S whose elements, (belonging already to another set A, distinct from S), would be sets that are not members of themselves.

The Axiom of Separation, as the name indicates, creates (an infinite number of) separate sets, one for each conceivable property, out of the original and thus avoids, allegedly, self-reference.

The apparent advantage of this axiom above my own analysis is that S can still be considered as a legitimate set, whereas with me it was demoted to a mere name or description.

The choice seems quite simple, doesn't it? Let us find out.

What is a set that is not a member of itself?
A set of dogs is itself not a dog, a set of mountains not a mountain. There are infinite examples of such sets. In fact, most sets will probably be sets that are not members of themselves.

Sets that are members of themselves are in fact much more mysterious, and seem to be defined only in a negative way: the set of all non-dogs is itself a non-dog; the set of all non-mountains is itself a non-mountain...

Let us look now at the most defining property: elements of S must be elements of A. So if a set is to be considered a set of dogs in S, it will have to be first a set of dogs in A.

As it turns out I do not think it is  possible to exclude the inheritance of the antinomy. Let us not forget that A is both a member of itself, and not a member of itself. We will not be able to choose between one of those two contradicting properties, and S will remain at best undefined. 

We cannot exclude A when it is not a member of itself, and neither can we when it is, because that would mean excluding all members of A, therefore not creating S. Unless you can exclude A without excluding its members? A fascinating possibility!

Of course, this objection rests on the assumption of an antinomy within A. But then, without this assumption, why would we need an Axiom of Separation?

I would not be surprised if I went wrong somewhere, and if I did, I cannot wait to know where.

The Liar Paradox (and other beasties)
Reductio and Wishful Thinking:

Enderton, in his excellent introduction "Elements of Set Theory" (1976) shows why, in his official opinion of expert of Set Theory, the set of all sets that do not belong to themselves cannot be a set and, most importantly, why the subset formed out of the elements of the suspicious set, is itself not a member of that set, and is therefore contradiction free.

Here the whole argumentation:

Theorem 2A There is no set to which every set belongs. 
Proof Let A be a set; we will construct a set not belonging to A. Let 
В = {x  ε A | x ф x}. 
We claim that В ф A. We have, by the construction of B, 
ε В iff ε А&ВфВ. 
If BεA, then this reduces to 
BεB iff ВфВ, 
which is impossible, since one side must be true and the other false. Hence 

[I had to replace the gibberish character with iff (if and only if). ф  means the negation of the epsilon symbol, that is " not a member of" or "does not belong to".]

This could be a nice example of a reductio were it not for a fundamental difference. The reductio argument assumes a premise which is shown to lead to a contradiction. In other words, the contradiction is created by the assumption even though it needs not be. When Euclid, as (one of) the first used this form of proof to prove the non-existence of a highest prime, and therefore of the infinity of primes, he did not create a contradiction by affirming: 

1) There is a highest prime. 

There is nothing self-contradictory in this assertion. Only, the argumentation shows that we necessarily arrive at the negation of this assertion: 

2) There is no highest prime.

And that is the end of it. (1) is taken out of the running once and for all. It has been proven to be a false assertion.

This is certainly not the case with 

3) The set B (whose members, belonging to A, do not belong to themselves) is not a member of A.

This proposition Is not proven ad reductio by

4) The set B (whose members, belonging to A, do not belong to themselves) is a member of A.

The proposition (4) is not taken out of the running since it has not been proven to be a false assertion.

Both (3) and (4) form both horns of the dilemma. Asserting one is the same as asserting the other. Affirming that it "is impossible, since one side must be true and the other false" is a pure declaration of faith that does not make the antinomy go away.

The Liar Paradox (and other beasties)
Reference and geometry
When Euclid affirmed that we can draw only one line through a point, he had not only understandably failed to anticipate non-Euclidean geometry, he also did not fully realize the meaning of his assertion. After all, how can we draw a triangle, or even a simple angle if both sides cannot not share one and the same point? But they do, don't they? Where is then the problem?
Well, in fact that would mean that you can draw at least two lines through a point, which would not be very Euclidean. Unless of course one line is considered as jumping over the point which has already been claimed by its sibling.

We know that solid objects cannot do that: you cannot have two different objects occupying the same space at the same time. That is why geometrical lines, just like geometrical points, have no spatial dimensions. Our drawings are mere pedagogical tools to help us understand abstract principles. Furthermore, engineers have learned since Antiquity how to translate those drawings into real constructions.

Let us now, for the sake of argument, consider both principles as equivalent:

A) you can draw only one line through a point
B) two solid objects cannot occupy the same space at the same time.

Suppose that these principles were hard-coded in reality, that is, even drawing more than one line through a single point was physically impossible. How would we draw a triangle?

Here is a possible method:

1) draw 2 vertical and parallel lines as close to each other as (physically) possible.
2) draw 2 oblique lines, each one from the top of the corresponding vertical line with any arbitrary inclination, not necessarily the same on both sides.
3) keep drawing oblique lines on both sides, with the same original inclination (s), keeping the same minimal distance with the outer lines as the distance between the vertical lines, starting each time from a lower point respectively on the vertical lines.
4) Draw a horizontal line, the base, to form a triangle containing the smaller triangles. You must have now have two large triangles separated by the minimal vertical, or horizontal, distance.
5) Draw a few parallel lines to the base between the base and the top.
6) you can now see that each line contains a different number of points.

This would be a physical refutation of the Cantorian principle that a subset can be put in a one to one correspondence with the larger set.
Also, the expression, 

C) the top of the triangle is a point

would have to be replaced by: 

D) the top of a triangle is formed by two points.

Let a djinn now suddenly reduce the minimum distance between the points in such a way that it would disappear from our perception. (C) would then become once again a very plausible assertion. But would it describe reality? For that matter, would (D) describe reality?
After all, the distance between two points may not be empty. Maybe the following expression would be more appropriate:

E) the top of a triangle is formed by three points.

But then why stop at three? Will CERN help us one day find the right expression?

This shows how hazardous it is to let language refer when it is obvious that even its users have no idea what they are talking about.

The Liar Paradox (and other beasties)
This is inspiring. But logic and phenomenologic claim proof of truths and falsehood,can also be found in between 'statements' after the modern greats in this area of thought from the 2nd WW. Say Heidegger's dasein, even Kant's noumena are full experiments in the scientific philosophical sense. Ethics is the underlying driving force to make this possible. So Pegasus is an example of this with no accomodation in the material sense but a truth in the phenomenological. This states that logic must be perceived beyond the mind in psychology to make it vast and open to philosophers.

The Liar Paradox (and other beasties)
Remark on Reference and Geometry:
I realize now that the Euclidean geometric rule concerns the drawing of one line through two points. Still, I do not think that it makes any difference for my argumentation. After all, when drawing a triangle you have to draw two different lines through each point.

The Liar Paradox (and other beasties)
A Djinn in Paradise
Imagine you are a powerful djinn who caught all natural numbers in a magic bucket. You want now to enumerate different objects to see what all the Cantorian commotion is about. You are especially interested in the different forms of infinity since you find the one in which you have been restrained all your life somewhat... restraining. The idea of a greater freedom is a powerful motivation, and you get to work.

You decide to investigate the infinitely small regions first, that would put that much more distance between you and the Big Boss, giving you more opportunity to expand your horizons. 

You follow Cantor's instructions and start putting all the reals (at least the part after the decimal point) between 0 and 1 in an infinite list, making use of the ingenious diagonalization method of this smart human.
After an infinite while you stop and ponder. It does seem like there can come no end to the list of reals between 0 and 1. Would it be possible for a canny djinn to hide between those reals after a major mischief?
You look at the infinite list for a long, infinite time until you finally notice the bucket from which you have been pulling all those numbers to draw those reals. 
You sit and ponder.
What would happen to the reals if this magic bucket ran empty? What if the Big Boss took it away while you were hiding in there? How would you ever come out of it?

The Liar Paradox (and other beasties)
Reference and Infinity
Fraenkel et al, "Foundations of Set Theory" (1973), after criticizing Dedekind's view of infinity because it led to the Russellean paradox, (any object we can think of can be put in a set, and there is a set of all objects we can think of) affirm the following: "From the axiomatic viewpoint there is no other way for securing infinite sets but postulating them." They then propose a few formulations of the Axiom of Infinity. I will leave the subtleties of those different formulas to those who can appreciate them and concentrate on the first one, which happens to be also very readable as far as I am concerned.

AXIOM OF INFINITY: There exists at least one set Z with the following properties 
(i) 0 ε z [the empty set is a member of Z]
(ii) if x ε Z, also {x}ε Z

Which of course is supposed to mean that whatever number you can think of, starting from the empty set, it will be a member of Z, as will the set containing this number:

0, x, {x}, {{x}}, {{{x}}}, {{{{x}}}}...

This infinite series is then understood as representing the infinite series of natural numbers.

This is how we are supposed to interpret the words and expressions making up the axiom of infinity. But what do these words and expressions really say?

1) Set Z exists
2) Z contains the empty set. But it is not altogether empty since
3) it can contain x and also the set containing x, {x}.

Where is Infinity in all that? All we have now is a set, whatever that is, with 0, x and {x}, whatever they may be. Methinks we are still a long way from infinity!

Let us see if we can instrumentalize this definition and turn it into a computer program [or shall I say app?]. Maybe Infinity will materialize somehow.

100 a=0; advance memory pointer
101 a=a+1; advance memory pointer
102 goto 101
103 end

Here is a fascinating puzzle: what happens when we read 102? You will agree with me that the instruction is quite clear. But you will also agree with me when I say that reading 102 is not the same as executing the instruction contained in it. If that were the case we would be unable to stop, just like a computer.
Apparently, neither writing nor reading this little app [sigh] creates anything remotely close to infinity. How about executing it?

Well, the computer will certainly produce a very long list of numbers. But as far as the machine is concerned, it is only executing one instruction at the time, whatever the instruction may be.

How about us? Can we say that the computer is creating an infinite series of natural numbers? Of course we can! And that, despite the fact that neither the computer program nor its source code contain any hint of infinity.

So, Fraenkel et al were certainly right when they said that we "postulate" infinity, even if they kept the illusion strong that we can somehow express this postulating into words or axioms.

[This does not make their efforts useless or meaningless. We need words to express and convey our thoughts, and even though words can never replace thoughts, they remain nonetheless indispensable. Also, there is obviously quite a difference, whatever that is, between words and sentences that help us understand and communicate clearly, and those which do not.]

It seems that not only are we not able to prove the existence of infinity, we cannot even express it into words (or axioms). Whatever we can say does not and cannot refer to infinity unless we put it there.

How about the word or concept of "infinity" itself? Doesn't it refer automatically to Infinity? Doesn't it just like 'apple' refer to something real or imagined?
Of course. Please remember that what is at stake here is the question of the empirical (or logical, fictional, mathematical, scientific) existence of the object in question. If we did not understand the meaning of "apple', then we would not be able to refer to it in the real world, or in a sentence in a dialogue or a book.
That is what makes meaningful a sentence like

"The boy ate the apple before William Tell could shoot."

Even if neither the boy, nor the apple, or for that matter, William Tell, ever existed.
But to say that "infinity", or in that case,"infinite" is somehow autological is, as far as I am concerned, just gibberish. "Infinite" is not infinite; neither the word nor the concept are infinite. Both are quite delimited and precise. What is infinite is what we imagine or think of in our head, not the signs, physical or mental, we use to express it with. 

So no, even 'Infinity' does not create Infinity, just like 'apple' does not create apples. We do, when we do not just find them in the real world.

The Liar Paradox (and other beasties)
Reference, Paradoxes and Logic

I have the impression that the following clarification would be far from superfluous.

Antinomies pose a challenge to logic. Intuitionists are more conscious of this fact than others since they seem ready to put into question Logic itself, or at least the tertium non datur or exclusion principle: a statement is either true or false, it cannot be both.
Non-intuitionists prefer to pretend there is nothing wrong with Logic, and that the culprits are the antinomies themselves. Once they have been banned, everything will return to normal.

But how do you ban antinomies? 
By declaring them non-logical, and therefore inadmissible in the Court of Logical Thought. 
Non-intuitionists govern by decree: Thou shalt not believe in antinomies nor aid them in their devilish endeavors!

I follow neither of those traditions since I consider antinomies not a logical problem but a linguistic one.
Let me try to put it as clearly as possible.
It is not a matter of formulation. Antinomies are not just false problems because of the ambiguities of natural language. What is at stake is our fundamental conception of language as a tool of expression. We supply the thoughts to sentences which, left to themselves, are just inert signs.

The Liar Paradox (and other beasties)
I hope my other entries answer somehow to your first objection that I did not deal with a formal paradox.
As far as Habermas is concerned, he is certainly not the only one, nor the first to speak of natural language as its own metalanguage. Concerning his conception of truth you might look up his work on communication, starting with his monograph published in a book with Niklas Luhmann in 1971 "Theorie der Gesellschaft oder Sozialtechnologie".

The Liar Paradox (and other beasties)
Reference and Syntax

Jerry Fodor is known for at least one steady conception in his very productive career: syntax takes care of itself. This Chomskyian principle is at the basis of the Language of Thought and was already part and parcel of modern philosophy of logic. Statements are true or false according to their form or syntax. Semantics are of course an indispensable part in thought processes, but, to paraphrase a modern (cyber) slogan, "Syntax ruleZ!".

I find it quite a strange claim when one realizes how long it takes for an individual to make sense of all the strange signs found in Logic books. [See below] 
If we take pre-school learning into consideration, as we should, I would say that we are looking at an average close to 20 years before a student can do anything else but parrot his teachers.
Logic as a faculty might be innate, but that is certainly not the case of Logic as a discipline.

If you start trying to learn Logic as an adult outside of the protected environment of a campus, you are confronted with all the different idiolects that different authors use as their own. A mysterious entity like the empty set appears to have at least three or four symbols at its disposition. The same holds even for such elementary concepts like equality. Logicians consider it as very versatile and distinguish, among other things, the case where a=b is supposed to mean that both a and b are the same object, from the esoteric possibility that only attributes or properties, whatever they may be, are involved.

You also have to interpret strange strings of symbols that, or so you are told, have only one interpretation. But when, in your naive assiduity, you try to do just that, you find out that the verbal explanations of the authors do not really make any sense to you. Or more precisely, that you find them sometimes meaningful and insightful, the problem being that they do not seem to make the interpretation rules any clearer. In fact, very often, you are unable to make any sense of the statement even though you understand all the individual symbols; the perfect definition of a culture-shock. You feel then compelled to take the writers at their words. 
Strings of symbols which have still not yielded their secrets get then an authoritative interpretation that trickles down to other formulas until you have been drilled to react appropriately to the right strings of symbols.
But how do you make other people conscious of this drilling effect? How can you make them understand that the language they have been using all their life is not as clear as they think it to be, even without taking Logic into consideration.
I knew a young man, a distant relative whom I met by chance, who was residing illegally in the Netherlands and I found him someone to teach him the language. His teacher came to me after a few weeks and complained that she found it impossible to explain to him the meaning of the verb "to be". Semitic languages do not use it in the present tense, only in the past or future. I would therefore say "I El Ouggouti", meaning "I am El Ouggouti". Which explains the famous "You Tarzan, me Jane" syndrome. A sentence which was, in Tarzan's mind, grammatically utterly correct.
Most people learn this peculiar use of the copula unconsciously, but not my relative, who was obviously over-thinking it.

This obstinate myth, logical symbols have only one interpretation [in its weaker form, logical symbols should have only one interpretation, and it is the task of Logicians to, slowly, realize this ideal] not only neglects the necessity to drill students in the use and comprehension of the symbols, it also denies a simple fact: the same symbols can apparently be interpreted very differently by different authors. Without it, progress would be practically impossible and confined to philosophical debates about axioms and primary assumptions [maybe that is already the case?]. Arguments would be tight-clad each time, unless human errors took place. After all, syntactic necessity would only permit one way of proceeding. 

Also, try to make sense of the axioms of Set Theory, in its many variants, without a thorough knowledge of the history that led to them. 
I would call a "naive Set Theory" a theory which would appeal to the intuition even of a layman, not some intellectual creation that needs years of study before you can even start to make sense of it.

You do not have to be a Benacerraf to admit that whatever the different models of Set Theory prove, they certainly do not explain how you ever learned to count.


I realize that Fodor means a "mental" syntax, not necessarily the rules of logic as they are found in textbooks. But since I cannot judge of such an elusive concept, I will limit myself to what an external syntax can be. Besides, the principles unveiled by Fodor are, as I see it, not so much those governing thoughts as their expression. For instance, we cannot express our thoughts to others, or even to ourselves, without dividing them into chunks and somehow connecting those same chunks. But it does not mean that we actually think this way. So, in "Language of Thought", I would put the accent on "Language", and not "Thought".
Anyway, I am far from convinced that Fodor has unveiled more than practical conditions to the expression of thought. He is still a long way from a syntax of thought, if such a thing can be said to exist.

Furthermore, as I tried to show in my entry "Logic, Mathematics and the Brain" in  The Brain: some problematic concepts,, even a simple statement as

A) "IF 1+1=2 AND 2+1=3 THEN 1+1+1=3"

cannot be properly translated into computer instructions, which are syntactic statements par excellence. This does not not bode well for a purely syntactic language that could replace natural language.


The Liar Paradox (and other beasties)
Reference, Logic and Syntax

When the modern logician replaces every concrete term by a variable, he is doing something that even the great Aristotle never thought of. The question is, would it have really mattered if "The Master", as he was called by the Ancient Arab Philosophers, had thought of it?

Imagine Aristotle saying something like:

1) All A's are B's
2) S is an A
3) S is a B.

I can just imagine the heated debates in the Ancient Greek academy:

student1 (whispering to his neighbor): what the ... is he talking about?
student2 (raising his hand and speaking at the same time): what is an "aye", and how do we know it is a bee?
student3 (standing up and throwing a lost look around): is 'Es' really a name?
student4 (to student1, whispering back): the old man has lost it.

The teacher raises his arm and the class reluctantly settles down.

11) All Athenians are Brave
21) Socrates is an Athenian
31) Socrates is Brave.

Student at the back: Yeah! He drank the poison without a twitch!
Many voices in the class: - Right on! 
- He's the man! 
- Socrates had balls!

[Now you know why Aristotle wrote so many books. He could not wait to get away from his classes.]

Let us say that after a while the students got used to the peculiar presentation of their teacher and even started friendly competitions about who could devise the best logic riddle. Here is one they came up with.

Student (full of pride and confidence): you guys will never guess this one!

12) All M's are M's
22) M is an M
32) M is M

After the protestations about the unfairness of the riddle had gradually quieted down, the studious assembly finally admitted its defeat and humbly asked for the solution which was then promptly given.

13) All Macedonians are Meek
23) Miko is a Macedonian
33) Miko is Meek.

What is even more peculiar than the humorous way students were learning to cope with the new strange language was that nobody really stood still but what it meant for the human faculty of thinking. All agreed that variables replace any arbitrary concrete term, as as such were much more apt to express the generality of an argument. In fact, even the quantifier 'all', or the article 'an', become superfluous in the new language.

14) A's are B's
24) S is B
34) S is A

One did not need to wonder anymore how the logician could speak of 'all' when he obviously never could know them 'all' [see again the same thread mentioned above, all entries concerning logic].

But that is not the most important aspect of all. The fundamental question remains the same: what makes us believe 1-3 in all their forms? And is there a difference between believing 1-3 and 14-34?

When we say

4) IF Athenians are Brave and Socrates is an Athenian THEN Socrates is Brave

5) IF A's are B's AND S is B THEN S is A

Is (5) fundamentally different from (4)?

After all, even Aristotle' students knew that the names were just dummies, place holders. Can we compare the historical passage (With I think Boole and Frege as pioneers) from (4) to (5) to that of rhetorical mathematics to modern notation (which started, according to some, in the 15th century and took more than 300 years to complete)?

I honestly do not think that the change brought progress to Logic the way it did for Mathematics. In fact, the formalization of Logic has, in my eyes, greatly contributed to its mystification. It became an esoteric discipline, just like mathematics. Both disciplines shared the evidence of primary intuitions, belief in the validity of a statement in the case of Logic, counting in the second case. These, let us call them faculties for lack of a better name, may without doubt be considered as universal. 

[there are some anthropological reports of isolated tribes that had only names for one, two and many. I have never been a fan of Anthropology which I think has always been, and still is, the colonialist version of sociology practiced by the White Man in dark continents. You will therefore pardon me my skepticism concerning the explanations given. Especially when you think that in those tribes it was very common to have many children, and to lose as many to disease. So, even if the people concerned lacked comparable words or numerals, their language had to be rich enough to speak of more than two, three or four, and less than a swarm. After all, as the anecdote goes, even crows can count until four!]

It is my deep conviction that those fundamental faculties keep being active at every level of Logic and Mathematics as disciplines. Which means that, however fancy the formulas may become, they are still no more than a form of belief in one case, and of counting in the other [geometry and related domains demanding more than counting].

The Liar Paradox (and other beasties)
Addendum to Dedekind's cut:

Can we account for all the points of a diagonal in a unity square?

Let us look at the method described above (the entry Reference and Geometry).

There is no reason to start at the top. We can draw parallel lines to the side starting from the base.
But before that, we have to make sure we are dealing with a square and its diagonal. By drawing the second diagonal we create four equal  triangles. The point on each half of a diagonal can be put on a one to one correspondence with the  points of the other sides respectively, taking into account the principle of one line for each point. 

This puts Dedekind's remarks that some points cannot be put in one-to-one correspondence with rational numbers in a new light: the only explanation is that the points only detectable with irrational numbers are in fact not located on the line itself, but are a product of his own formulas.

Anyway, a very interesting question is, looking at

a2 < D < ( a+ 1)2
[I have replaced the lambda sign with a]
and knowing that "this cut is produced by no rational number", what does this formula refer to?

The Liar Paradox (and other beasties)
Pythagoras and Perspective

[Ye be warned who enter here: the following offers a pure view of the mind, unhindered and unspoiled by any trivial knowledge of worldly affairs.]

The fact that the diagonal of a unity square is itself different from 1 has posed quite a conundrum not only to philosophers and mathematicians, but also to science in general. How come we are unable to measure exactly the diagonal using the same unit as with the sides?
Even irrational numbers are not really a satisfying solution, except in a practical sense. After all, the diagonal has a fixed, precise and delimited length which can be measured using other unities of length.
It seems like the distance between two points differs with the perspective. But why should it surprise us? After all atoms are not perfectly circular, and if we can go by their big brothers, the stars and planets, they generally tend to the elliptical, as do their orbits. The axial distance of a heavenly object will therefore differ according to the perspective of the observer.
Understood this way, what would be surprising is if there were no difference at all between measures taken from one perspective and those taken from another.

The Liar Paradox (and other beasties)
Where are my 2.3 kids?
Statistics is obviously a domain that will always make use of irrational numbers. There is therefore hope for the other numbers too. The metaphysical implications will still disappear from mathematical statements. Nobody believes seriously in 2.3 kids. We all know it is, like  Gauss would say, a mere façon de parler.

The Liar Paradox (and other beasties)
Cantor's Logic (1)

In "Contribution to the founding of the theory of Transfinite Numbers" we are dealt a nice sample of Cantor's way of "argumenting" about numbers. [The symbol ~ stands for 'equivalent to'.]

(9) M~M.

The second M is a set, the first M, with two bars on top, is the cardinal number of the set M, its power, or more simply, the number of elements belonging to the set M.
What could (9) possibly mean?
Until now M was a "normal set, an aggregate of units which in our mind were considered as one. There were also relations possible between a set M and a set N, relations which depended on what we call nowadays bijections, a one to one relationship from one element of M to one element of N, and vice versa.
There was therefore no doubt that we were dealing with (abstract) objects on one hand, and a number on the other.

Now, as it would seem, Cantor is interested in proving that

=   =
M=N is the same as M~N, and inversely. 

that we can go from a statement about the equivalence of cardinality to a statement about the equivalence of sets and inversely. In his mind, that could only mean that the distinction between a set and its cardinality could then be ignored without consequences.
But is that so? I am afraid that the analysis of this Cantorian position will take a long time, so allow me to approach it gradually.

"Greater" and "Less"
To explain this relationship Cantor presents two conditions:
a)There is no part of M which is equivalent to N,
b) There is a part N1 of N, such that N1~M

A very ingenious description of the relation between M and N. 
It is not immediately obvious with the first condition, but if we look closely at (b), we realize it assumes perceiving that N contains a part equivalent to M.

M_____________________ N
xx___________________ xxy
x____________________ x
xx___________________ xxy

I used y to bring out the common x's to M and N. They are of course not necessary, we would see the pattern even if only x's were used.
The first condition seems like the second face of the same coin: M, and a fortiori a part of M, could never be equivalent to N since M contains no y's.

(a) and (b) sound like the verbal report of a non-verbal perception. They are how we would explain to someone else what we saw if we did not have the concepts of "greater" and "less"... And If we wanted to sound as mysterious and obscure as a Greek Oracle.
Indeed, if we look at the first statement more closely we realize that it is plain riddle-speak for what we are seeing when looking at two sets one of which larger than the other one.
The same can easily be said of the second statement or condition, even though the first one was in itself sufficient to convey the message that N is larger than M. 
(b) in other words is completely superfluous.

The rationalization, or more precisely, the verbalization [in a non-law-enforcing sense] of a perception process is even more pronounced in the following paragraph (par.484 P.89). 

[the numbering is mine]:
"[1] Further, the equivalence of M and N, and thus the equality of a and b is excluded;
[2] for if we had M~N, we would have, because N1~M, the equivalence N1~N,
[3] and then, because M~N, there would exist a part M1 of M such that M1~M, and therefore we should have M1~N;
[4] and this contradicts the condition (a)."

[1] could not be more obvious since we had already seen that one set is larger than the other. 
Starting from [2] we are dealt a reductio again: M~N, which is in flagrant contradiction with what we had already concluded on basis of our perception. [3] confirms the contradiction and [4]expresses it.

A nice sample of unnecessary statements under the guise of an argumentation. In fact, all that those statements say is: what we see cannot be explained otherwise, because every assumption that we see something else leads to a contradiction with what we see. Duh!

Maybe those statements could be useful in proving the greater than relationship between two sets when we have no direct perception of these sets. Let us look at them from this perspective.

a) There is no part of M, which is equivalent to N.

First we must make sure we understand what is meant with this statement. It is not that M cannot be put in a one to one correspondence with a part of N. That is always possible even when N is empty. Don't logicians consider the empty set as a member of each set?
No, what is important here is that no part of M, including the whole of M, can be put in a one to one correspondence with the whole of N.
In other words, there will always be elements of N which are not part of the bijection. Which makes N automatically greater than M.

Here again, condition (a) is sufficient for expressing N>m (or M
What remains is the not unimportant fact that with this statement we make it possible for a person who is not looking at M and N, to reach the same conclusion. The problem is that what we are asking of him is simply to be able to interpret correctly (a). That certainly demands a modicum of intelligence, but none of logic since the latter expects at least two statements before it even deigns enter the action; and he does not need (b) anymore since he already knows enough.
We are compelled to draw the same conclusion as before: (a) is a very round-about way of saying that M
b) There is a part N1 of N, such that N1~M

We are here confronted with a possible ambiguity: can a part of N be the whole of N? We know that every set is a subset of itself, but Cantor apparently means here part (Bestandteil) in a more traditional way. And so shall we.
This decision makes it easier to understand (b) as simply saying: 

b1) a part of N is equivalent to the whole of M, 

or, what amounts to the same thing, 

b2) the whole of M is equivalent to a part of N.

In other words, any way we look at it, M(b) is also a self-sufficient condition for the > and < relationship.

You will allow me to skip the renewed analysis of the following argumentation, it would only entail boring repetitions, except that instead of contradicting our perception, the so-called proof contradicts what is already known.
But isn't that the essence of a reductio?
Not really, unless you want to turn a respectable logical form into a play of words.
At each stage, Cantor uses a cryptic statement that contradicts directly the known facts, and then concludes that he has proven this fact. While all he has been doing is contradict the original fact in words and not in deeds (or logical steps).

To go back to Euclid's example once again, Euclid showed that the highest prime led to its negation. All Cantor does is say the same thing with different words.

Cantor has apparently made school since we found the same misuse of the reductio argument by Enderton [see above, the entry Reductio and Wishful Thinking].


The Liar Paradox (and other beasties)
Cantor's Logic (2)

"We will call by the name "power" or "cardinal number " of M the general concept which, by means
of our active faculty of thought, arises from the aggregate M when we make abstraction of the
nature of its various elements m and of the order in which they are given."

Cardinal number is here a concept, and not a set.

This become less obvious with the following lines:
"Since every single element m, if we abstract from its nature, becomes a "unit," the cardinal number M is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate M."

Here a cardinal number starts looking like the set itself, or the object of which it is supposed to represent the number of elements.

The jump becomes a giant leap when Cantor declares:
"between the elements of M and the different units of its cardinal number M a reciprocally univocal (or bi-univocal) relation of correspondence subsists."

It sounds like we do not have a single number anymore, but as many as there are members in a set. And that brings us to the mysterious statement (9):

"For, as we saw, M grows, so to speak, out of [cardinal] M [with two upper bars] in such a way that from every element m of N a special unit of M arises. Thus we can say that

(9) M ~ M." [the second M is also cardinal M, with two upper bars.]

Understood this way, each element or unit of the set becomes a unit of the cardinal number, and therefore '1'. Every set is a set of 1's, which makes it easy to calculate the final cardinal numbers, and the results of all known (arithmetic) operations. 
This certainly seems to make sense since we had stripped the elements of a set from any specific property other than their membership of the set. We then might as well call them 'units' or '1's', both expressions being equivalent.

The problem of course is the concept of a cardinal number on one hand, and the individual units or 1's which, when added up (if the operation in question is addition), yield that cardinal number.

It seems to me that, even if we consider the cardinal number as a set, we have in one case a set with only one number, the cardinal number, and in the other case a set containing different units or 1's (which, taken together, represent said cardinal member).

(9) is therefore very problematic, to say the least.

The Liar Paradox (and other beasties)
Corrections to Cantor's Logic (1)

The text editor interprets the "greater than" and "less than" signs as the start of special computer instructions, like TAB or END OF LINE (slash n).
This resulted in truncated sentences.
[I use italics for the sentences as they appeared and bold fonts for the corrections.]

"Here again, condition (a) is sufficient for expressing N>m (or M" 

should  end with 

"(or M less than M)
[you understand if I do not use the symbol itself.]

"We are compelled to draw the same conclusion as before: (a) is a very round-about way of saying that M"

should be

"We are compelled to draw the same conclusion as before: (a) is a very round-about way of saying that M Less Than N (or N Greater Than M). But instead of presenting the fact on a platter, we make the hearer (or reader) work for it."

[as you see, the rest of the paragraph following the special symbol was deleted!]

"In other words, any way we look at it, M(b) is also a self-sufficient condition for the > and < relationship."

are in fact two sentences:

- "In other words, any way we look at it, M less than N (or N greater than M)."

The following sentence is supposed to stand alone:

- "(b) is also a self-sufficient condition for the > and < relationship."

The Liar Paradox (and other beasties)
Cantor' Logic (3): The Addition and Multiplication of Powers

The identification of the cardinal number with the elements of a set pays its dividends right away. Like I mentioned before, it makes the introduction of arithmetic (and other mathematical) operations seem quite natural.

For instance the sum of two sets poses no problem at all. One only needs to remember that the cardinal numbers M-double bar and N-double bar are equivalent to the numbers a and b.
And as we know:
If we add a number c we get
a+(b+c)= (a+b)+c.

These rules, and others similar to these, seem self-explanatory. Until we ask ourselves which cardinal numbers we are speaking about.
When we say a+b, do we mean the cardinal number denoted by a single number, like 5 or 7, or the different units making up this cardinal number?

Take the schema we used before:

M________________ N
xx_______________ xxy
x________________ x
xx_______________ xxy

How do we add M to N without the use of numerals, be they that of natural or cardinal numbers, and whatever the difference might be?

I honestly would not know, at least not where large numbers are concerned, and I do not think Cantor did either!

Natural numbers seem to surreptitiously hitch a ride by every definition of mathematical operations on sets.

The Liar Paradox (and other beasties)
It's easy to deal with antinomies in set theory. The axiom of infinity is false. Much ado about nothing.

The Liar Paradox (and other beasties)
Cantor-Bernstein Theorem: a real or apparent problem?

There are many proofs of this theorem, that Cantor called the Equivalence Proposition (Satz), but of which he never published a proof. In fact, Arie Hinkis' book "Proofs of the Cantor-Bernstein Theorem" (2013), is more than 400 pages long, and he does not even cover all of them. So you will understand if I choose the simplest one of them. To be honest, the most proofs I would not even understand!

I found Borel's version ("Leçons sur la Theorie des Fonctions", appendix I, 1898), of Berstein's proof reasonably readable (that goes also for Kertesz' presentation in his "Einfuhrung in Die Transfinite Algebra", 1900) but if push comes to shove, I think I will work with my own version!

As you know, the whole point is to prove that two sets, A and B, are equivalent when they are known to be equivalent to one subset of the other, respectively A' and B'.

A~B' AND B~A'--> A~B

One very obvious method is to enumerate the different possibilities (there are 4 of them) concerning the relationship between a set and its corresponding subset. Then it appears that only one possibility is of interest for us, the case where, as Borel put it, A has the same power as B', and B as A'.
Hinkis (p.60 and further) is quite critical of Borel and finds the enumeration of the different possibilities superfluous. I personally think it rather sets the argument in the right perspective.  It certainly helped me in rephrasing the theorem, hopefully without betraying its essence.

1) If a subset A' of one set A is equivalent to a set B
2) and a subset B' of the set B is equivalent to the set A
3) then the two sets A and B are themselves equivalent.

Here again we are confronted with what I would call the Oracle-style or the Tongue of Riddles. The obscure formulation creates, in my eyes, the false impression of a problem that has to be solved, the empty conviction of the necessity of a solution that has to be proven logically and/or mathematically.

But what happens if we solve the linguistic riddle first?

10) if one part of A, including the whole of A, is equal to the whole of B,
20) and if one part of B, including the whole of B, is equal to the whole of A,
30) then the whole of A is equal to the whole of B.

(10-30) can be simplified even further:

11) If the whole of A is equal to the whole of B,
21) and if the whole of B is equal to the whole of A
31) then the whole of A is equal to the whole of B.

And now I wonder: what have all those proofs been proving for more than hundred years?

[Hinkis: "The naive reaction of someone educated in college set theory, is that CBT for sets of infinite numbers must be trivial. Our proof shows that this is not the case in the context of early Cantorian set theory." (p.20)
That may be so, but the way the theorem can be rephrased is not refuted by considerations of infinite sets and their peculiar relation to their own subsets. In fact, assuming that subsets of infinite sets have the same power as their parents makes the simplification of the theorem (from 10-30 to 11-31) even more acceptable. This makes Hinkis' speculation that Schroeder committed suicide because he was ashamed of his mistake even more tragic!]

The Liar Paradox (and other beasties)
Cantor' Logic (4): The Exponentiation of Powers

After having secured addition and multiplication, Cantor turns to a crucial part in his analysis, the creation of ever larger sets via exponentiation of powers. 
He starts by defining the concept of function in a set.

"The element of M bound up with n is, in a way, a one-valued function of n, and may be denoted by f(n); is is called a "covering function of n". The corresponding covering of N will be called f(N)."

Cantor calls f(n) a "covering function of N with M".

What is worth our attention is the identification of the element m with the function that connects it to a corresponding element n.
The cautious "in a way" should be taken very seriously.

The following lines create their own problem though.

"The totality of different coverings of N with M forms a definite aggregate with the elements f(N) ;
we call it the "covering-aggregate (Belegungsmenge) of N with M " and denote it by (N | M).

(2) (N|M) = {f(N)}."

What is meant here is the totality of f(N)'s, and that is a to the b'th power, ab, with a and b, as before, being the cardinal number of respectively N and M.

The question is, does this totality even exist (in the mathematical or logical sense)? After all, once you have defined two sets by a function, you have once and for all determined the nature and properties of the elements of both sets.

Let us look at a simple example.

M____f(N)____ N
4____ f(2)____ 2
9____ f(3)____ 3
25___ f(5)____ 5

I wonder if it would be possible to define a function completely different from "the square of n" for those two sets. 

So, in a way, ab  only makes sense if we consider the elements of f(N) as indeterminate units which can be represented by any function at the same time.
Still, that would demand an extra factor, the number of possible functions, which we do not have. No, what is meant here appears to be  a pure numerical relationship between N and M. Each element of N is assumed to have M possible values. But that is quite arbitrary, isn't it?

This is obviously a case where the identification of a set with its cardinal number is very problematic. 
ab may make sense mathematically, but it lacks any logical consistency.

And I am not even stopping to wonder whether f(N) can be legitimately considered as a set. After all, it looks very much like the name of the relation between N and M: "M is the square of N". 
Dividing this statement in its different occurrences:
"4 is the square of 2;
 9 is the square of 3;
25 is the square of 5",

does not (necessarily) turn this relation into a set distinct from N and M. But then, in the so-called "Naive Set Theory", anything can be a set. I should remark however that even though such a set does not, as far as I can see, create any antinomy, its legitimacy remains questionable.

When we consider that exponentiation of powers is the introduction to fundamental propositions concerning transfinite numbers, then it becomes more important than ever to evaluate this step as critically as possible.

The Liar Paradox (and other beasties)
The Pitfalls of Infinity: Dedekind's cut

In "Essays on the Theory of  Numbers" Dedekind wants to prove the continuity of the straight line but he finds himself limited by the nature of the Pythagorean dilemma: there is no common unity between the diagonal and the sides of a unity square. There is therefore, according to him, no possibility of accounting for all the points in a line using rational numbers. The only solution to save continuity, is by introducing irrational numbers.
"Since [...] it can be easily shown that there are infinitely many lengths which are incommensurable with the unit of length, we may affirm: The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals."

This affirmation has apparently a great convincing potential since it has never been put in doubt, even though it is far from obvious. After all, a straight line is only incommensurable with other lines under certain conditions. Any line will be commensurable at least with itself. It is therefore not evident, nor proven, that "The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals."

Apparently Dedekind has based all his argumentation on a wrong assumption. I wonder what that means for his analysis?
The reader will understand that as a non-mathematician I need to be overly cautious in anything I can say on this subject. I will certainly take my time. That is the least I can do.

Anyway, until now, neither physical arguments (Physics assumes discrete quanta of matter) nor geometrical considerations (the gaps in a straight line are not necessarily present) seem to support his approach.

The Liar Paradox (and other beasties)
The Pitfalls of Infinity: Calculus

This would look like a mathematical issue, but what is at stake here are our primary assumptions and how we think the world fundamentally is. In a word, what is at stake here is our logic of space.

Newton's fluxions and Leibniz' infinitesimals have changed the way we look at the world by according to a philosophical view the legitimacy of science. Like Russel indicated in his introduction to a new edition of "Principles of Mathematics" (1902): "The doctrines of Pythagoras, which began with arithmetical mysticism, influenced all subsequent philosophy and mathematics more profoundly than is generally realized." In fact, it shaped the way science still views matter and space. Not even Quantum Theory with all its theoretical and technological successes has succeeded in dislodging this "arithmetical mysticism" from its pedestal. You would think that the concept of discrete quanta would once and for all take care of this irrational attitude and the so aptly called irrational numbers. Nothing is less true.

Let us say you want to calculate the surface of an irregular shape like a leaf or the outer borders of a country.
Nowadays you could just scan the image at any available resolution and you would have the "exact" distance between any two points. The problem would still be how to translate those pairs of numbers in 2D figures.

Imagine a wonderful, magical future where machines will be able to measure accurately the size of different atoms, electrons, and even elementary particles. They will still have to choose an arbitrary unity of measure, and each choice will have as a consequence that other quantities will only be known by approximation. They could of course have huge tables comparing different elements with each other, a database which would permit the substitution of measuring units at will. I dare wager that our descendants would then still be dreaming of a time when science would be so advanced as to allow the use of an absolute unity of measure, just like we do. 

Will these not-so-advanced-as-they-wish scientists of the future still need irrational numbers for their physical calculations? After all, irrational numbers were born in a time where the "ainek mezanek" principle was in power ["you eye (is) your scale"].
Why would you need those numbers when you could accurately name the number of (any arbitrary kind of) atoms, plus, if necessary, the electrons and elementary particles involved?

Because you can never be accurate enough? Let's look at this demand more closely.
It is of course easily conceivable that any geometrical figure will cut through any physical element at an arbitrary point. However accurate science can be, there will always be, in the case of a square for instance, elements which just touch the line, either from the inside or the outside, plus all the cases in between. Let us call this the "Ball-Out-Problem" or BOP. [Everybody is doing it, why wouldn't I invent my own cool acronyms?].
Any referee will tell you that whatever your decision is, it will always make some people very unhappy.

There is reason for optimism though.
If irrational numbers turn out not to be indispensable, will that mean that their offspring, complex, transcendental and imaginary children will get irremediably aborted? A very sad day for their fans, but many a future student would heave a sigh of relief.
Here is to hoping that mathematics will become simpler instead of ever more complicated. Heaven itself could become overcrowded if things keep going the way they are.

Even if things ended up the way I described it, Infinity would still remain a legitimate metaphysical problem. Nevertheless, we would have then at least dealt with the illusion that it can be brought down to mathematical formulas.

The Liar Paradox (and other beasties)
"Thank you for the fish"

I am sure you have seen a documentary of whales, dolphins or other sea hunters working together to encircle a school of smaller fish.
Imagine yourself a lone hunter/fisher, armed with a long net, but this time you are not hunting sardines but atoms and other imponderables.
Let us assume that each point on the perimeter you want circled, where the prey is waiting in fear, is an atom, or any other physical unit. You wouldn't need to worry about formulas for surfaces any more, neither make use of the venerable method of exhaustion, nor appeal to modern integral calculus. All you would need to do is draw a line between each pair of opposite points, respecting the one line per point principle (lets us call it the Point-Line-Point or PLP, to avoid embarrassing alliterations). The length of each line will also immediately indicate the number of units, and calculating the surface of an irregular plane, or its volume, will be as easy as adding the values of all the lines together, even after taking into account the ways atoms are placed to avoid the Pythagorean dilemma [see above Pythagoras and Perspective].
A future scanner should be able to give accurate measures of any geometrical figure, without having to worry about decimal points or rounding figures.

Sigh, I wish Geometry was that easy in high school. But then, knowing the teachers, they would have made us learn the boring stuff anyway.

The Liar Paradox (and other beasties)
Reply to Eray Ozkural
6:17 PM 1/12/2016 CET

The fact that you have published two articles (which I would be very interested in reading) makes you into a "professional" and gives you therefore the right to make such an incredibly naive statement.

If I attempted the same, my words would irremediably be lost in the cyber void. With any luck, you will read theses words, today or simply later, the nice editors willing, and whoever they may be. To them I express my deep gratitude even if I do not agree with all their rules... or rulings.

2:03 AM 1/13/2016
edit: I had reached my quota of public messages. I should not have made any corrections ["wants" instead of "want", "magical" instead of "magic"].

P.s: The negative effects of written statements, just like erotic pictures on social media, are very difficult to counter, even years after the fact. Since you have a professional reputation to lose, I would advise you to be more circumspect with your comments. At least motivate your sneer.
Furthermore, I never claimed that the Axiom of Infinity was false.

The Liar Paradox (and other beasties)
Cantor's Logic (5.1): The Finite Cardinal Numbers

After having given us a taste of what we can expect with the exponentiation of powers, Cantor judges the time ripe to lay down the foundations of his number systems, starting with the most elementary numbers.

The number "1":

"To a single thing e0, if we subsume it under the
concept of an aggregate E0 = (e0), corresponds as
cardinal number what we call "one" and denote by
1 ; we have
(1) 1 = E0 ." [cardinal number with two upper bars]

This is a very interesting definition of '1", the mother [or father? It does look like a phallic symbol] of all unities.

First, we have "a single thing" which everybody would just call '1", or 'one thing'. Everybody that is but Cantor.
Second, we need a (concept of a) set to put that single thing into.
Third, once we have that set, and the thing in it, we are allowed to call the cardinal number of that set '1'.

What was the cardinal number of a set again? Allow me to quote myself from the first part of this study:

"each element or unit of the set becomes a unit of the cardinal number, and therefore '1'. Every set is a set of 1's".

I must admit that the use of '1' was an extrapolation from my side. Cantor did not actually use the expression. 

Still, how are we to understand the statements he made a little bit earlier on in the text?

"Of fundamental importance is the theorem that two aggregates M and N have the same cardinal number if, and only if, they are equivalent : thus,
                                   =  =
(7) from M~N we get M=N,
and         =  =
(8) from M=N we get M~N."

How could a cardinal number be anything else but a natural number applied to (the elements of) a set?
Are we really dealing with a new concept, or merely with the technical designation of a familiar concept? 
If you are speaking of objects in general you may use the expression 'natural numbers', but when dealing with sets and their elements, we prefer you to use the technical term 'cardinal number'?

Or could it be that I am underestimating Cantor's geniality, that I am missing a distinction so subtle, but oh! So fundamental between the concept of a cardinal number and that of a mere (natural) number?
We can after all see that two small sets are equivalent, and for larger ones, we can always use the one-to-one correspondence to establish which one is greater or less than the other. Wasn't that what I myself showed in the first part where neither the concept or expression of natural number, and for that matter, nor the concept of cardinal number was used to explain the "Greater Than" and "Less Than" relationship?
Such primitive notions as the so-called "subitizing" or the one-to-one-correspondence method may certainly be considered as the forerunners of the concept of number, they could hardly replace it. The concept of "cardinal number" is much too advanced to be anything like those primordial beginnings.
It remains very strange though that the concept of natural number is considered to be born out of those same atavistic schema's that in Cantor's analysis gave rise to that of cardinal number.

A last possibility, as far as I can see, would be that cardinal numbers, unlike, natural numbers, still have to earn their numerals, or titles so to speak. The fact that both kinds of numbers make use of the same numerals would be nothing more but a matter of expediency and economy of symbols.
The problem with such a view is that it consecrates the distinction between the two groups without explaining it. In fact, it legitimizes the distinction, making us falsely believe that we already know of its existence and that we are okay with it.

We see here the same m.o as in the previous crime scenes: Cantor gives the impression that he is defining something new, while in fact he does no more than reformulate in his own obscure words what was already clearly known.

What we have to grant him is that he did not explicitly use '1' until now. Nevertheless, we can hardly say that the use of the concept or numeral '1' comes as a surprise.
This will become even more evident with the definition of 2, 3 and the rest of the numbers, whatever their nationality.

The number "2"

"Let us now unite with E0 another thing e1, and
call the union-aggregate E1, so that ,

(2) E1 =(E0, e1) = (e0, e1).

The cardinal number of E1 is called "two" and is
denoted by 2 :
(3) 2 = E1."

Okay then, we have 1, as element of a set E0, and we somehow bind it with another thing. Without being too pedantic let's try to do justice to Cantor's intentions and choose a charitable interpretation whenever possible.

(2) states that the union of a set with an object [(E0, e1)], is the same as the union of two objects [(e0, e1)].
In other words, we do not have to bother with the distinction between set and object at all since any combination will give us the same cardinal number.

I am bound to be charitable, but I still need to point to the ambiguity of (e0, e1) and choose the most favorable interpretation: 
(e0, e1) can be understood as simply a (spatial) arrangement of two objects, or it can be seen as a set which is the result of the union of two objects.
In the first case, speaking of cardinal number would be inappropriate since this kind of numbers has only been defined in the context of sets. When dealing with a group of objects, natural numbers seem the way to go.
We will consider Cantor justified in this case in his use of the concept of cardinal numbers and we should not give to much attention to the mixed usage of objects and sets, especially since later models of Set Theory will generally abandon the concept of object altogether and refer exclusively to sets.

Still, the main problem remains: how do I understand the distinction between a cardinal and a natural 2? If 2, or should I say '2', is to be more than a numeral that just happens to look and sound as the familiar '2', then there must be, once again, a way to distinguish one from the other. And if there is not, my question, as to the meaningfulness of the concept of cardinal number altogether, would be entirely justified.

The question becomes even more crucial when we realize that without the concept of natural number the passage from 1 to 2 would remain inexplicable.
After all, the only thing that the cardinal approach can show for its efforts are two isolated facts: the naming of a certain set as '1', and another as '2'.
In fact, the more cardinal numbers Cantor creates, the more urgent it becomes to assume a "number sense" (as Dehaene would say) which would run implicitly through the whole creation process. 

When summing up this process by

"E2 = (E1, e2), E3 = (E2, e3),...,"

and then declaring :

"The use which we here make of these numbers as suffixes is justified by the fact that a number is only used as a suffix when it has been defined as a cardinal number."

Cantor is asserting something he certainly did not prove: the meaningfulness of the concept of cardinal number, and its right to supplant the concept of natural numbers is far from evident, not to say utterly unjustified.

That is particularly obvious when we look closely at the following definition:
       =    =
"(6) Ev=Ev-1+1;"

"+1" (or -1), is a concept that is unaccounted for in the definition of cardinal numbers. It belongs exclusively to the number sense. Without the latter (6) would be no more than a name or description and the appendix '+1' would be a literal expression without any numerical meaning.
In fact, at each step of the creation process of cardinal numbers, Cantor takes the next natural number implicitly and sneaks it in to "create" yet another cardinal number.

It may be not so flagrant in the case of small numbers as in the expression already mentioned:
"E2 = (E1, e2)"

It becomes more than obvious if we let E2 be equivalent to a large set whose elements could only be counted to make sense of the expression. 

However we look at it, Cantor gives us no reason to consider cardinal numbers as in anyway different from natural numbers. The whole process looks like a superfluous, and historically probably unrealistic, duplication of the genesis of numbers.
Let us hope he will be able to make sense of them later on.

The Liar Paradox (and other beasties)
Cantor's Logic (5.2): The Finite Cardinal Numbers (continued)

[I strongly suggest to keep Cantor's text close at hand. Paragraph [489] of the critical edition, and par.5 in the text. P.97 and further.] 

Having "explained" the genesis of cardinal numbers Cantor turns to more advanced considerations.

Cantor first poses the following three theorems:
1) any cardinal number is different from its successor and predecessor;
2) any cardinal number is greater that its predecessor and less than its successor;
3) there can be no additional cardinal number between two consecutive cardinal numbers.

He then proposes to prove those theorems in two phases, first by two additional theorem D and E, second by more advanced proofs, or, as he says, "rigid proofs", of the same D and E.
First let me remark that it seems that D and E both say the same thing, the only difference being that the implicit assumption of M as "being of equal power with none of its parts" is made explicit by the presentation of N as a finite set.
Of course, the validity of D depends on the validity of the principle involved, that a set which is not equal to any of its parts is said to be finite, otherwise infinite. Charity demands of us that we concede this principle, as we will concede the validity of both the axioms of Well-Ordering and of Choice.
Let us look at the proof of D more closely.

Proof of D

Here is what D says:

"D. If M is an aggregate such that it is of equal power with none of its parts, then the aggregate (M, e), which arises from M by the addition of a single new element e, has the same property of being of equal power with none of its parts."

In other words, adding an object to a finite set will create a new finite set. The proof will be considered given if M+1, or in this case, (M,e) will not be equivalent to any of its parts just as M was not.

We are immediately confronted with a puzzle raised by the assumptions that (a) "The aggregate N contains e as element"... Or (b) "The aggregate N does not contain e as element."
How are we supposed to interpret these statements?
The object e represents a unity, and M is itself a set containing elements which are indistinguishable from each other. Once e has been added to M it necessarily vanishes in the crowd of unities.
The axiom of Choice has no opportunity to be put to work, there is after all nothing to choose between identical elements.
Both assumptions under respectively a) and b) are therefore meaningless.

Let us say we feel particularly charitable and decide to see beyond this hurdle.

First part of the Proof of D

We have 

1) (M, e) [the set created by the union of a set M, and an object or element e]

and we want to prove that, just like M, (M,e) is not equivalent to one of its parts.
To do that we will make use of N as a part of M. The aim therefore is to prove that (M,e)#N.

Let us consider first the case where e is also a member of N.
if e belongs to N then 

2) N=(M1, e) 
[M1 being a part of N, e an additional unity or object, the whole expression the union of both]

Cantor reminds us of the possibility or the absolute freedom expressed at the beginning of the article, to rearrange the elements of two pairs according to one's own rules [and which will be considered as falling under the Axiom of Choice by Zermelo and others].  We can therefore consider, in this case, the first e of (M, e) equivalent to the second e of (M1, e) - which is not so surprising since they both symbolize a unity- but also the correspondence of M and M1.
This makes us then realize that "this contradicts the supposition that M is not equivalent to its part M1".

I must say that this a really strange argumentation. Conform to my charitable attitude, I will grant Cantor the right or freedom to make M and M1 correspond to each other, my serious doubts notwithstanding. 
But the fact that it leads to a contradiction is not in itself a proof of its falsity [see the entries above, Reductio and Wishful Thinking, as well as the first part of this series].
After all, M could be, against all odds and expectations, an infinite set, in which case it would be equivalent to one of its own parts!

Because Cantor has not proven that M and M1, or M and N, could not be equivalent, he has no right, however charitable we may feel, to declare them as such.
This should not really surprise us very much since all he has done is, once again, play with words!

This becomes even more obvious if we still want to go further with our interpretation of those few lines and consider any contradiction as a valid reason to reject an assumption.
We could then wonder what the exact relation of N with M is. Are we to assume that N is a proper part of M, that M greater than N, or do we also have to take into account the possibility that M=N? 
Which possibility is the leading principle, if any, in Cantor's argumentation? Let us just keep this question in mind for the time being.

The subsection (a) can also be understood as based on the Well-Ordering Principle that states, loosely speaking, when two sets are such that (a,b) and (c, b) then we can conclude that a and c are equivalent. And since in this case a and c represent respectively M and M1 subset of N subset of M, it would mean that M is equivalent to one of its parts, which would be in contradiction with the original assumption.

The problem with this conclusion does not necessarily reside with the Well-Ordering Principle which we have decided to consider as valid. It has all to do with the relationship N and M.
If N is a proper part of M, then M1, as subset of N, can, by definition, never be equivalent to M. 
Remark that we arrive to this conclusion simply by clarifying what is meant by "part of". If "part of" always meant "proper part of", and "subset" always meant "# or =" [or vice versa], then no statement of this kind would ever be ambiguous and we would always immediately know whether a set, or element, belongs to another set or does not.

If N can be equivalent to M, then we have to consider again both possibilities, M1=N or M1 less than N.
In the case of M1=N AND N=M, then it is obvious that both M1 and N will be equivalent to M.
This is a perfectly legitimate possibility, were it not that it contradicts the initial assumption of M not equivalent to any of its parts. It would seem therefore that the leading principle is N as proper part of of M.

That takes care as well of the second the case of M1#N and N=M since it also is based on the contradictory assumption that M is equal to one of its parts.
Since M1 can only be less than N [N=(M1,e)], and N is not allowed to be equivalent to M, M1 will be always less than M.

In all cases, making explicit what is meant by "part of"" is sufficient to clear any ambiguity.
Also, the use of e is definitely misleading [the element e is never really needed to analyze the relation M and N], in that it opens the door to the Well-ordering principle which then seems, undeservedly, to legitimize the conclusion as one pertaining to a proof, instead of a linguistic clarification of concepts.

Most importantly, because of the "linguistic imperative" Cantor cannot claim to have proven that a finite set M, once an element is added to it, remains finite! He can of course appeal to our intuition and beliefs, but he cannot prove it.

Next we will look at the case where e does not belong to N.

The Liar Paradox (and other beasties)
Cantor's Logic (5.3): The Finite Cardinal Numbers (continued)

e does not belong to N

"(b) The part N of (M, e) does not contain e as element, so that N is either M or a part of M. In the law of correspondence between (M, e) and N, which lies at the basis of our supposition, to the element e of the former let the element f of the latter correspond. Let N = (M1, f) ; then the aggregate M is put in a reciprocally univocal relation with M1. But M1 is a part of N and hence of M. So here too M would be equivalent to one of its parts, and this is contrary to the supposition."

This looks suspiciously like a repetition of moves. The same curious argument is used as by the condition that e belongs to N, or that N=(M1, e). There Cantor used whatever implicit rules he trusted on, namely what was to become the axioms of Well-Ordering and that of Choice. That allowed him to consider, against all odds, M1 as equivalent to M. He uses in this second part the same strategy.
First, he reformulates N, to which e does not belong anymore, to
N=(M1, f); f being the equivalent of e in (m, e).
Having done that, he simply assumes M~M1, and declares it as a contradiction, and therefore as a proof that (M, e) is not equal to any of its parts.
The fact that M1 could not possibly be equivalent to M remains apparently unseen or ignored.
Why M1 cannot be equivalent to M?
M1 is a part of N which is itself a part of M. N can be therefore at most equivalent to M, but since that is not allowed, it will always be less than M.
M1 is necessarily less than N, since N=(M1, e). 

A final remarks that holds for the whole of the proof of D. 
Cantor starts with (M,e) but considers in his conclusions only M, ignoring e altogether. We must not forget that the aim was to prove that (M,e), when M was finite (not equivalent to any of its parts), was itself still finite. 
I have the strong impression that Cantor himself got confused by his conception of "part of". Instead of considering (M,e) as a totality, he slips and starts looking at the M of (M, e) as the new totality. That would explain his formulation "here too M would be equivalent to one of its parts, and this is contrary to the supposition".
He should have first assigned P=(M,e) as the new value of M; then tried to show that P could not possibly be equivalent to one of its parts.
As it is, he has been comparing not (M,e) or P to N and M1, but M to N and M1. The problem is that there is no reason why N could not be equivalent to M, or to P-e. And since M1 itself could then be equivalent to N, it could also be equivalent to M, but this time without contradiction.
In my own analysis I have tried to take into account Cantor's original intention, that of proving that the whole of (M,e), therefore P, is equal to none of its parts. 
So, however we look at it, Cantor has not proven that (M,e), just like M, is not equal to one of its parts, because he got confused and considered M itself both as a whole [as in M], and as a part [as in (M,e)]. He forgot his own formulation of D in which he set to prove that "the aggregate (M,e)... has the same property of being equal with none of its parts." (M,e), not M alone which was already assumed as not being equal to any of its parts.
The reason for this confusion lies, I think, in the use of e or an equivalent of it in all formulas. Because of it the distinction between a part of (M,e) and a part of M was lost.

The Liar Paradox (and other beasties)
The Myth of Mathematical Induction (aka M.I aka Mission Impossible)

Moshe Machover' "Set Theory, Logic and their Limitations" (1996) is a very readable book even if it had a very strange effect on me at some time. When I read in p.16:
[I have replaced the less than sign by LT]

"The relation LT obeys the trichotomy: for any numbers m and n, exactly one of the following three holds: 
m LT n or m = n or n LT m."

I felt just like one of Aristotle'students and my first reactions were then:
- who the... is Norm?
- is Norn really a name?

The shock was fortunately short-lived and I enjoyed his clear and simple exposition of mathematical induction.

Here is a verbatim quote.

"We shall prove that, for all n, 
(*) 0 + 1 + 2 +••• +n= n(n + l)/2.

Define the property P by stipulating that Pn iff (*) holds for n. We 
show by weak induction that All nPn. 

Basis. For n = 0 the sum on the left-hand side reduces to 0, and the 
value of the right-hand side is 0. Thus P0. 

Induction step. Let n be any number such that Pn; thus our induction 
hypothesis is that (*) holds for this n. Then 
0 + 1 + 2 + • • • + n + (n + 1) = n(n + 1)/2 + (n + 1) by ind. hyp., 
= (n + 1)(n/2 + 1) 
= (n + 1)(n + 2)/2. 
(The last two steps consist of simple algebraic manipulation.) Thus 
from the induction hypothesis we have deduced that 
0+1 + 2+--- + (n + 1) = (n + 1)(n + 2)/2. 
This equation says that P(n + 1) - it is the same as (*), but with n + 1 
in place of n. So we have shown that Pn => P(n + 1)."

With all due respect to the author, and all mathematicians,  the whole argumentation could be summed up as follows:

single case = squiggles
arbitrary case = squiggles
all cases = squiggles

The whole argumentation turns around the meaning of the arbitrary case. If we are convinced that it is of the same type as the single case, we will be more easily convinced to accept it as a general rule.

What does 'arbitrary' mean?
Answer: I will take a random example [just another word for arbitrary] and you will see that squiggles can be applied to it.

Once you have done that you will have convinced me that the first case, as well as the second, the one you have called by the name "arbitrary" or "random", both deserve the squiggles. 
But you want more of me. You want me to give you a signed, blank check for all future cases looking like those two cases. I am a cautious old man who has been conned many times in the past, so I ask myself: why should I?
"arbitrary" or "random", you say, means that I could put any name in its place. But I already knew that when I believed that
"single case"= squiggles

Imagine I tell you:

1) Mohamed is a king 

and you acquiesce. [I have always wanted to use that word. Thank you Philpapers!]
And then I say

10) Hachem is a king

And, to my horror and indignation, you dissent. [see previous remark].

I think it through for a while (I'm old and slow), and then come up with

20) Mohamed is a man

which you readily accept as true, just like you accept

21) Hachem is a man.

Then I ask you. "Would you grant me a blank check for the 1... statements as well as for the 2's? "
"Only for the 2's", you retort [...].

Why? Because not all men are kings while all men are men?
Just like all single cases are squiggles?
My philosophy teacher in the last year of high school (I suppose you could call it senior year) loved to repeat this statement: "l'habitude commence la premiere fois", habit starts the first time.

I think it was very deep and insightful, and completely wasted on high school kids.

I would like to honor his memory by applying it to beliefs and knowledge in general.
Every time we say something of a single thing, it is like we said it of all things like it. The same way, anytime we see a thing behave a certain way, we expect all things like it to behave the same way.
[This is the deep reason why racism and prejudice are so hard to counter.]
The so-called arbitrary or random cases are, in our brain/mind, indistinguishable from other singular cases:
[see Logic and the Brain in The Brain: some problematic concepts]

To say 

3) M is a K 

is not fundamentally different from saying that 

1) Mohamed is a king.

Hume was in a way quite wrong when he assumed that we needed repetition or habit to arrive at our conclusions. A single case is enough.

Logicians care deeply about the illusion that the use of variables somehow justifies the (possibility of) validity of statements. 
They are wrong. A statement containing only variables is just as singular as one containing proper names. And the latter is just as general as the former.

Throughout his history Man did not need formal Logic to turn singular affirmations in general rules, nor the other way around.

What does this mean for mathematical induction? Does these considerations turn it into an invalid form of argumentation?
Certainly not. As long as we realize that we cannot "prove" a rule on the basis of one or more cases. We just give ourselves a reason, by appealing to the similarity of the new cases with what we already know, to believe that we should treat them the same way.

This would certainly be trivial and justly considered by mathematicians as what the Dutch call a "ver van mijn bed show" (a show far from my bed, or something that does not concern or touch me), were it not that Mathematical Induction is very often misused in philosophical and metaphysical arguments.

I hope to show that Cantor is guilty as charged,

The Liar Paradox (and other beasties)
I hold a computer science PhD, and in my research I have a very strong commitment to foundations of mathematics, logic and probability theory. I work on general purpose AI theory which is founded on algorithmic information theory. Those are just some philosophy papers you saw. What I do is designing general purpose machine learning algorithms, and writing theoretical articles on various aspects of general-purpose AI. Attacking my reputation like that is amusing but completely ad hominem, and is improper in an academic forum.

I said that the axiom of infinity should be considered false. I didn't think you said it is false. On the contrary, I think now, that you probably don't know that all semantic antinomies in set theory would be resolved if we rejected the axiom of infinity. (Assuming everyone here knows what a semantic antinomy/paradox means) Which is a repetition but must be said. Most people naively think that axiom of choice is the culprit (this is not so formally). 

I believe that we are speaking much ado about nothing when we pretend that there is anything meaningful about antinomies in set theory, which assumes that somehow, ZFC theory, is semantically complete or sound. I hope this clarifies my position on this, I am of course, like some constructivists, a finitist of sorts. We finitists usually don't think that the set R "exists", for instance. I instead prefer the set of computable reals.

The Liar Paradox (and other beasties)
Cantor's Logic (5.4): The Finite Cardinal Numbers (continued)

Proof of E

This is what E says:

"E. If N is an aggregate with the finite cardinal number v, and N1 is any part of N, the cardinal number of N1 is equal to one of the preceding numbers I, 2, 3, . . ., v1."

As I mentioned before, E fundamentally says the same as D, using numeral variables instead of the general concept of finite set.

Cantor did not prove D, will he be able to prove E? 

Let me first remark that it looks like a mathematical induction in reverse. We would expect Cantor to try and prove that the next element also has the desired property, instead of that he looks backward. If the current element has property p, than the preceding element has a corresponding property.
It does seem a little bit strange because it involves elements which we are supposed already to have encountered before we got to the current element. Mathematical induction has normally something of a prediction. Here, we are trying to understand the past. A historical project as it were.
Such an approach is not necessarily invalid, but it certainly needs more justification that the usual one.

You can then imagine my surprise reading the first lines of the proof.

"Proof of E: We will suppose the correctness of the theorem up to a certain v and then conclude its validity for the number v+ I which immediately follows..."

Cantor apparently chooses anyhow for a traditional approach.
Or not?

"We start from the aggregate Ev = (e0, e1, . . ., ev) as an aggregate with the cardinal number v+1. If the theorem is true for this aggregate, its truth for any other aggregate with the same cardinal number v + 1 follows at once by [par.1 which says that two sets with the same cardinal number are equivalent]."

Cantor apparently wants to prove three different statements:

i) if an element has a cardinal number v, then preceding elements will all have cardinal numbers less than v;
ii) if an element has a cardinal number v, then posterior elements will all have cardinal numbers more than v;
iii) if (1) or (2) are true of a set with cardinal number v, it will be true of all sets with cardinal number v.

All three objectives are certainly very closely related  (not to say strangely obvious), still, we will have to be vigilant and distinguish one case from the other to judge of the validity of the argumentation.

We are starting therefore with:

1) Ev = (e0, e1, . . ., ev)
2) E' any part of E.'

Cantor considers three possibilities, a, b and c.

(a) "E' does not contain ev as element, then E is either Ev-1 or a part of Ev-1, and so has as cardinal number either v or one of the numbers 1,2,3,...,v-1, because we supposed our theorem true for the aggregate Ev-1, with the cardinal number v."

This is obviously supposed to be a proof of the first objective (i).
But proof is such a big word for such a triviality that Cantor does not dare use it at all. In fact, all three possibilities are simply presented without a heading or title. Still, they all fall under the main heading "Proof of E".

I will try to show at the end of this entry that the use of the concept "cardinal number" - which, if I may remind the reader, has never been shown as being anything else but the technical designation of "natural number" when applied to sets- is quite misleading as to the meaningfulness of all the statement where this expression appears. 

In the case of (a), let us see what the effect would be if we changed "cardinal number" to "natural number", or any such equivalent expression like "the number of elements".

(a') "E' does not contain ev as element, then E is either Ev-1 or a part of Ev-1, and so has as its number of elements either v or one of the numbers 1,2,3,...,v-1, because we supposed our theorem true for the aggregate Ev-1, with the number of elements v."

(a') can be expressed even more simply: 

(a'') If the number of elements of E' is less than or equal to v, then this number will be either v or any number between 1 and v-1. 

You immediately realize that we do not even need the reference to an additional theorem. The first part of the statement is self-sufficient, and also clear and evident enough.

"(b) E' consists of the single element ev then
E'=1." [cardinal E'=1]

We would not think that any commentary is necessary for such an obvious statement. Until we realize that such a statement is full of meanings. It confirms the image of elements as 1's, and that we are therefore dealing with disguised natural numbers.
Also, curiously, this tautology is not among the statements which Cantor had set out to prove so that it is left hanging in the air.

(c) E' consists of ev and an aggregate E", so that E' = (E", ev). E" is a part of Ev-1 and has therefore by supposition as cardinal number one of the numbers 1, 2, 3, . . .,v-1. But now cardinal E'= cardinal E"+1, and thus the cardinal number of E' is one of the numbers 2, 3, . . ., v."

Let us use the same substitution principle on (c) as we did on (a):

(c') E' consists of ev and an aggregate E", so that E' = (E", ev). E" is a part of Ev-1 and has therefore by supposition as the number of its elements one of the numbers 1, 2, 3, . . .,v-1. But now the natural number E'= natural number E"+1, and thus the natural number of E' is one of the numbers 2, 3, . . ., v."

Does that sound like a problem that is really in need of a solution?
(a) is also a very obscure reformulation of (i), just like (c)!
So far, (ii) and (iii) are missing altogether from this analysis. Which should not worry us too much seen as how close all three statements are to each other. But it still remains a very sloppy presentation.

In conclusion: did Cantor prove E? However charitable we may feel, I do not think that we could concede that point to him, or even allow him the benefit of the doubt.
Next we will look at the so-called "rigid proofs" of the various theorems concerning finite numbers.
Please be prepared for a very disappointing end to this little adventure 

The Liar Paradox (and other beasties)
Cantor's Logic (5.5): The Finite Cardinal Numbers (End)

Rigid Proofs:
Cantor does not use the expression a second time, which is as well, since the additional "proofs' he presents of the theorems A, B and C are not really very convincing.

Let me remind you of the theorems we are dealing with, but this time using A, B and C instead of 1, 2 and 3 like I did in the second part.

A) any cardinal number is different from its successor and predecessor;
B) any cardinal number is greater than its predecessor and less than its successor;
C) there can be no additional cardinal number between two consecutive cardinal numbers.

Proof of A:
First, Cantor reminds us that this theorem, as far as he is concerned, has been proven by theorem D. I, of course, beg to differ.
Second, that the set of 1 (0,1) is not a part of itself is pretty obvious since neither 0 nor 1 are equivalent to (0,1). It does not therefore any extended proof but is recognized at once.
Third, u being less than v, the cardinal number of any v will be different from that of the corresponding u: cardinal v-1 # cardinal u-1.

Notice how this so called proof is in fact a mere reformulation of has been said before. Cantor does not add anything new to his prior analysis.

Proof of B:
The same strategy is followed here as by B, the proof consisting in no more than a reformulation of the theorem and what he has allegedly achieved with D and E.

Proof of C: more of the same.
Apparently the so-called rigid proofs are more a recapitulation of the different steps than anything else

Why would Cantor follow such a superfluous strategy? I will not try and guess what was on his mind when he wrote those lines. What is obvious to me though is the role cardinal numbers play in this strategy. To make it clearer to the reader I propose to apply the substitution rule used earlier:
Let us substitute "natural number" or "number of elements" for every occurrence of "cardinal number" or "power" and see how the Cantorian theorems fare.

A) any natural number is different from its successor and predecessor;
B) any natural number is greater than its predecessor and less than its successor;
C) there can be no additional natural number between two consecutive natural numbers.
D. If M is an aggregate such that it has the same number of elements as none of its parts, then the aggregate (M, e), which arises from M by the addition of a single new element e, has the same property of having the same number of elements as none of its parts.
E. If N is an aggregate with the finite number of elements v, and N1 is any part of N, the number of elements of N1 is equal to one of the preceding numbers 1, 2, 3, . . ., v-1.
F. If K is any aggregate of different finite number of elements, there is one, Kv amongst them which is smaller than the rest, and therefore the smallest of all.
G. Every aggregate K = {k} of different finite number of elements can be brought into the form of a series 
K = (k1,k2,K3,...)
such that
k1 LT k2 LT k3,...

Reading the re-formulated theorems we then realize why Cantor really needed a concept of number distinct from "natural number". My first reaction would be to say that he could hardly come up with such trivial truths and expect to be taken seriously! History shows that such an interpretation is most probably too simple, if not to say naive.
In "Was sind und was sollen die Zahlen?", a decade earlier, in 1888, (translated as "The Nature and Meaning of Numbers". the second part of "Essays on the Theory of Numbers") Dedekind also tried to account for an abstract genesis of numbers. He did not use any extraneous concept though, dealing directly with natural numbers.
This has also been the approach followed by mathematicians after Cantor. Dedekind's way was apparently more attractive than Cantor's indirect and uselessly complicated strategy.
An interesting later development is analyzed in detail by Piergiorgio Odifreddi in his seminal work, translated from Italian, "Classic Recursion Theory" of 1989/1992.

Cantor's legacy is not my direct objective though. What I am interested in is how he built his system, and less how it was interpreted by others later on. That is why I will keep very close to the text itself and give not as much attention to later developments for the time being as some would may be wish.The almost exclusive attention to the antinomies of "Naive Set Theory" has overshadowed the other weaknesses in Cantors system, making it seem like the paradoxes were its only problem. I believe that is not the case.

In summary: I do not consider the introduction of cardinal numbers as a happy choice. In fact, it has contributed greatly to the needless obscurity of the analysis and the end results that Cantor can show for his troubles are quite meager.
It would be useless to dwell on the many weaknesses in his analysis, pointing at them once along the way is more than sufficient. 
The main point, as far as I am concerned, is that Cantor did not successfully prove that adding elements to a finite system yields a final system in return, and that is a very serious shortcoming in his system. He starts the construction of transfinite numbers on a very weak basis.

The Liar Paradox (and other beasties)
Cantor's Logic (6.0): The Transfinite Cardinal Numbers 

Cantor's logic is nowhere so evident than when he is presenting his favorite toys, the Transfinite Numbers. I find it extremely puzzling that he could convince not only himself, but generations of brilliant minds of the validity of his approach. Since ignoring, or worse, belittling his charisma would only raise hairs and shields I propose to, once again, adopt a charitable attitude towards his thinking in the hope to find out what his secrets are that made him so credible against all odds. I am aware that Cantor had a really difficult time for many years, but the fact that he overcame most, if not all of his detractors, is certainly worth a deeper look. I propose to look for his secrets in his writings, especially in the Contributions, and not in the attraction his idea might have had on professional mathematicians. Enough has been written on the subject and adding my own voice to the choir would have little effect.
Still, a charitable attitude does not in itself imply some form of apologetism and I certainly will not shy away from any critical necessity.

Since this is not a scholarly study of Cantor's oeuvre, but rather the analysis in depth of both articles almost line by line, I will confine myself to the text itself with very few "sorties" to other works.
What is important is not what Cantor might have thought on the subject in general, or what others think he might have thought, but the exact analysis of what he has said in the Contributions, and therefore to gauge the extent to which he succeeded in proving his points, or at least making them sound plausible to the ears of the public.
My Leitmotiv will be: what did Cantor prove exactly in the Contributions (Beiträge)?

Aleph Zero:
It is the first transfinite number that Cantor presents to us, and as such of primordial importance.

"The first example of a transfinite aggregate is given by the totality of finite cardinal numbers v; we call its cardinal number [...] Aleph-zero"

The most familiar objection was to the concept of an actual Infinity, which is obviously present in this definition. I propose to concede Cantor the right to speak about it as an existing phenomenon. What I am interested in are not so much his metaphysical choices than the way he defends them. Forbidding him to believe in the actual infinite, for whatever religious or metaphysical reasons, has not worked, and I see no reason to advocate a hopeless strategy.

So, what is this "totality of finite cardinal numbers"? Or rather, what could it, and what could it not be?

Assuming an actual infinity, we picture, just like the mischievous djinn did, all the finite cardinal numbers present before our eyes. 
Now put all those numbers into a single set.
If you think about it, that is quite a conundrum. We have all the infinite numbers on one hand, and an act of thought that turns them all in a unity. Instead of an infinite list of numbers, we have a single set, and in that set we have a single number that, just like the famous ring of Sauron, binds them all. Since we are dealing with actual infinity we are entitled to think of that ultimate number as something as real as the infinite list of numbers itself.

I propose, for typographical reasons to substitute the aleph zero symbol with Z0, and that of cardinal number with subscript c

"(1)  Z0= vc"

will therefore represent the expression "aleph zero is equal to the cardinal number of v".

Being a transfinite number does really strange things to you.
Apparently adding 1 to Z0 will not change it, it will still be Z0:

"(2) Z0+1 = Z0."

The result is in conformity with our intuition and mathematical conventions:


And that is no more than what (2) in fact tells us. How could we not agree?
And since any finite number u, however large, will always be smaller than u+1, we can diffidently claim that Z0 is larger than any u.

"(3) Z0 greater than u."

If you take away a finite number, however large, from Z0, you will still have Z0. The same way, any transfinite subset of Z0 will be equal to Z0.

As you can see, substituting the infinity sign for Z0 would make all these statements sound much more familiar and acceptable.

Why would Gauss, and all the others who opposed so virulently Cantor at the beginning, would consider Infinity as a façon de parler? Instead of asking him let us just imagine his answer. What is important is not academic or historical authenticity, but understanding.

Lets us go back to the idea that whatever you add or subtract from the infinite, you will still be left with the infinite.
Rationally, we would all, I think, agree that even infinity +1 is different from infinity tout court. However negligible, we have the feeling that the +1 does have a meaning, otherwise why say it?
Similar ideas can be found in the ancient riddle of the straw that broke the camel's back, or of the drop falling in the ocean. Obviously, in human terms, a straw or a drop would not make any difference to the state of the the camel's back or to the ocean. But imagine erosion without that single drop of water dripping from the mountain, or evaporating in the air. Suddenly, the effect of a single drop takes on a whole new meaning.

It is therefore very human of us, and of course of all mathematicians, to say that infinity+1, or even infinity + infinity is still infinity, but that is certainly not how the universe and evolution work. The Parmenidian perspective, man as the measure of all things, sounds suddenly very provincial.

This larger perspective makes it at least plausible that talking about numbers larger than infinity may not be as senseless as we first thought. But is that what Cantor had in mind with his transfinite numbers?

Maybe so, but before we follow him on that path, let us look a little bit critically at how he got so far. Let us take another look at

"(2) Z0+1 = Z0."

I consider (2) as the cornerstone of Cantor's edifice. Showing that it is not a valid statement will probably bring the whole building down.

What are the arguments supporting (2) in the Contributions?

"if to the aggregate {v} is added a new element e0, the union-aggregate ({v}, e ) is equivalent to the original aggregate {v}."

This affirmation is nothing else but (2) expressed in words, and explains therefore nothing. The following is what it is all about:

"to the element e0 of the first corresponds the element 1 of the second, and to the
element v of the first corresponds the element v+ 1 of the other."

how are we to understand this strange and totally unexpected assertion?

Notice that e0 is an extra element, and represent therefore a unity, not an empty set, nor 0. After all, it would be very curious to add 0 to a set and claim that the set has somehow become bigger. Also, there can never be more than an empty set, or a zero, in a set, and we are , maybe, entitled to think that v, as the set of all cardinal numbers, like any set, has the empty set as a part of itself. Nonetheless, we must not forget that 0 has not been defined as a cardinal number. The first cardinal number that Cantor defined was 1.
Still, however we look at it, we can reliably say:
[I will use i's for my own statement, to distinguish them from quotes.]

i) e0=1

The question therefore is, how could the first element of the new set ({v},e0) be e0 while it was the last element added? And why should we care, since they are all unities?
But that is exactly the problem, isn't it? The elements of {v} are not unities, they are cardinal numbers 1,2,3,...v.
It seems like the zero element is made part of a set only when it is convenient.
Cantor is, at least that is the impression I get, relying on the presence of the index 0 of e0 to suggest that the first element of {v}, which would be 1, corresponds to the element 0 of ({v},e0). But such a suggestion is completely unjustified. Why would we make 1 of one set correspond to 0 of the other set?
Unless we are somehow counting the elements, the one to one correspondence being subservient to that objective.

Here is the difference between both processes:

One to one correspondence as a method of checking whether two sets are equivalent, or whether one is larger or less than the other: assuming that the order does not matter, we take each element of one set and place it in front of one element of the other until we have had them all.

One to one correspondence as a method of counting: in this case, we will take one element of one set, and place it in front of 1, then another in front of 2... Until we have counted them all. In such a case, assuming the set to be counted starts with 0 and ends with v, the counting set will start with 1 and end with v+1.

Cantor's argumentation is definitely a sleigh of hand: a counting set will always show as larger than the set counted, or as is the case here, as being as large as the set +1, or ({v},e0).

A fundamental law of transfinite numbers seems therefore to be illegal.

ii) Z0+1=Z0 is false!

The legitimacy of Aleph Zero is seriously in doubt.

The Liar Paradox (and other beasties)
Cantor's Logic (6.1): The Transfinite Cardinal Numbers (Continued)

Cantor's Aleph Zero is a deficient concept for at least two reasons:

1) The difference between a finite and an infinite set has not been proven. We cannot prove, even if our intuition says so, that adding an element to a finite site does necessarily create a finite set in return. The passage from the Finite to Infinity remains therefore a mystery.
This is even more so if we accept the definition, which I do not, of an infinite set as a set that is equal to one its parts. Adding more elements could possibly trigger the metamorphosis of a finite set into an infinite one. We have no way to know.
Except for our intuition, we must rely on Mathematical Induction to make plausible the idea that what is valid for a number n also holds for n+1. But right where it comes to the transition from the Finite to the Infinite, Mathematical Induction starts to look like a Mission Impossible, without the big screen heroes to save the day. Mathematical Induction as a method and as a principle declares Infinity impossible: there will always be a n+1 successor of n.
The impossibility to prove Infinity has as a curious consequence that everything is in principle measurable, and even countable. I will come back to this controversial statement later on.

2) The main principle on which the concept of Aleph Zero rests has been shown to be invalid. The idea that adding any number of elements to Z0 still yields Z0 as a result rests on the confusion of one-to-one correspondence as a checking method on one hand, and as a counting method of the other.
In the light of all of the above, what can Aleph Zero possibly mean?

No set is equal to one of its parts:
We need to tackle the issue of an infinite set being equal to one of its parts. The fact that  Z0+1#Zdoes not invalidate this principle.
Let us look again, in more depth, at the argument Cantor presented in the Contributions so far.
I have already treated of the argument D in the entry 5.2 of this series. I have shown that Cantor got somehow confused and ended up proving something wrong, namely that a subset N could not be equal to a part of M of (M,e), where he actually intended to prove that it could not be equal to a part M of (M,e).
I propose to rectify this simple confusion under the Charity principle, and see what the value of the argument is when it has been interpreted correctly. We already know, or at least strongly believe and suspect that a finite set will always be greater that any of its parts (not including itself), and therefore different from any of them. 
What we are interested in is the alleged failure of this principle when it comes to infinite sets.
To throw a light on this curious phenomenon we will have to leave the safe ground we have covered in the Contributions until now. We will look not at the complex method of diagonalization, for that we will need more experience with the Cantorian texts and logic, but at the much simpler proof that

(on)even members are as numerous as the set of natural numbers.
To avoid the fallacy unveiled in the analysis of Aleph Zero, I will postulate the absence of zero or the empty set in all sets involved.
Taking the even numbers as an example, the argument of the equivalence of this set with that of all the natural numbers is in fact quite simple: starting with the first even number, we assign to it the number 1, and to each following even number, the following natural number. 
To the infinite series

2 4 6 ... corresponds the infinite series
1 2 3.... 

a) This argument rests on an unspoken sense of Infinity. We cannot imagine it stopping somewhere, each even number 2n can be followed by 2(n+1), to which will correspond n+1.
We are dealing therefore with a circular argument assuming the existence of what we wish to prove: Infinity.

b) The argument is quasi self-referential: 
There is only one magical bucket out of which all numbers are pulled. 
This does not automatically dispel the illusion that we are dealing with a single infinite series of numbers which somehow bites its own start, or points at its different parts. 
The question is: how do you point at your right index finger with your right hand?
The answer is childishly simple: you need a second right hand.
That is what makes the argument quasi, or even pseudo self-referential. There are an infinity of 1's, 2's, 3's.... which we can use to point at other 1's, 2's, 3's...

As a conclusion we can say that the equivalence of the set of (on)even numbers and that of the natural numbers is a fallacy. The only thing the argument proves is that the process of pointing at any subset of the natural numbers is an infinite process because the natural numbers themselves are infinite. We cannot therefore prove infinity by referring to the infinite subsets, because we would not have infinite parts if we did not have infinity in the first place.
Still, an infinite part would seem to remain a very useful indication of the infinite character of a set. I certainly would not have any objection if mathematicians looked at it this way. As long as they realize that you may approach infinity through different ways. But different ways do not different infinities make.

Back to our problem: are (on)even numbers as numerous as natural numbers or not?
Think about it for a second. How could you answer this question by the affirmative, or the negative for that matter, without first putting a limit to both sets?
Maybe now the temptation to consider infinity as actual will be clearer. 

But Actual Infinity creates its own paradoxes.
Actual infinity is conceptually equivalent to a precise limit (not in the mathematical sense). Let us call it v like Cantor so often does.
Our problem then looks likes this:

2 4 6 8 ....... 2v
1 2 3 4 ....... v

Now my question: how do you know you are dealing with an infinite set and not a large finite one?
My answer: you don't.
Actual Infinity is conceptually indistinguishable from the Finite.  You may say that v represents the cardinal number of Aleph Zero or whatever mysterious set you can think of when thinking about Infinity. But you cannot prove it. Finite and Infinite become both empty labels indistinguishable from each other.

So, to answer your question: you may, if you want, consider the set of (on)even numbers as equinumerous (one of Frege's favorite terms) with the set of natural numbers. No lives depend on it, because you could say the opposite with as much right.
That is the mystery and beauty of Infinity: it is infinitely patient with our (intellectual) shortcomings.

That brings to my last remark, for now, on Aleph Zero. It is, as we know, defined as the totality of finite cardinal numbers. Not as the finite totality of finite cardinal numbers. Finite numbers are infinite in their own right. Let us not forget that

The Liar Paradox (and other beasties)
Diagonalization and its problems

Is there a difference between actual infinity in one dimension, and on multiple dimensions?

When we consider the set of even numbers (or uneven, rational...) as countable we are assuming two things.
1) We can use the set of counting numbers in an orderly way, without having to turn back on our tracks.
2) infinity can be actual

Let us take a closer look at the second assumption, and take all reals, starting from 1 (we can always get back to those starting with zero later on). Is is possible to conceive, independently of the question of countability, of those numbers as an actual infinity? If it is then we have an infinite series of infinite series which can be all considered as actual. 
After all, it would be very difficult to justify an exception rule for natural numbers since each one of those infinite series is itself made out of natural numbers.
We can consider all those actually infinite series of reals as forming a two-dimensional matrix, but that would be for our convenience only. There is no reason the number of dimensions could not itself be infinite.
Take now any one of those actually infinite series, it will be, as already stated, in principle countable. Let an infinite number of djinns each put one series in a one to one correspondence with the set of natural numbers.
This infinite number of djinns will then then be able to count all reals, won't they?

[Are you worried about the Axiom of choice, how would each each djinn know which series to take? Let us say that each djinn just picks the first series close at hands and looks for a corner where to work in quiet. There may be some duplicates of series, but who cares? The djinns have their numbers working for them too.]

Does your head hurt? Maybe that is because Cantor was right also. Humans cannot count the reals the way they can count even numbers. That does not mean that so-called uncountable sets are in themselves uncountable. Their uncountabilility is a property of our knowledge, not of their nature.

The Liar Paradox (and other beasties)
Mathematical induction works, actually. The AGI (abbrv. for Artificial General Intelligence, aka general-purpose machine learning) system I developed did solve such problems in 2010, and there are many experiments by others. We can apparently induce both functions and mathematical formulae.The final system will be able to induce much better than (limited) human mathematicians can. Carnap's theory has been vindicated, and we actually are using pretty much the same theory as Carnap proposed. IOW, I think that you are wrong on all accounts, but this is not something to worry about. We are refuting Platonism wholesale, refuting the vapid criticisms of induction like the so-called "problem of induction" or the white raven paradox, was just the first salvo. Of course, AGI researchers are actual mathematicians with advanced degrees in mathematical sciences, so I think that we do know better.

The Liar Paradox (and other beasties)
Reply to Eray Ozkural
I commend you for your academic achievements, and I wish you had stated your arguments the first time. However, formulating your position is not justifying it. I have much sympathy for the constructivist approach, even though I have to admit not to know it in all its details [I know even less about finitism]. I would be very interested in the reasons of your claim that the axiom of infinity is false. I honestly do not know how rejecting it would resolve all antinomies in set theory, but I am willing to learn if you are willing to teach [please do not just refer me to books elsewhere, I can find them myself when the time is ripe].
Also, I never pretended to defend Set Theory, nor attack its legitimacy in general. My first approach was how we understand language and how that understanding influences how we understand reference. From there I landed on the subject of Cantor and Infinity. A subject which I had dug into a few years ago without any definitive results. This was my opportunity to pick up the thread again.
To be clear, I am not interested in a metaphysical discussion of Infinity, but in the arguments that mathematicians, especially Cantor, have devised.
Last but not least, I am surprised that as a finitist you did not comment on my view about the Pythagorean dilemma. Did it skip your attention, or does it not fit in with what you understand under finitism? I am eager to know.

The Liar Paradox (and other beasties)
[I speak in 6.1 of "(on)even numbers". You can consider the expression meaning "(un)even numbers" but said with a Cockney accent. The truth is that the Dutch spelling was making fun of me.]

Cantor's Logic (6.2): The Transfinite Cardinal Numbers (Continued)
The following analysis will sound very pedantic to the layman, and probably even to many mathematicians and logicians. I would wholeheartedly agree with them and would gladly forsake it, especially since the modern models of Set Theory use only sets, having done away with objects almost completely.
Still, the very artificial character of this analysis can be a good explanation as to why, eventually, the distinction between objects and sets was dropped.
It also shows which problems the concepts of objects and sets had to face before, and hopefully also show whether the decision to drop objects out of the theory has necessarily solved all the problems born from this relationship. (which I do not think is the case, since turning aleph zero into a set would not solve the issues treated below.]
For my part, I will concentrate here mostly on how the distinction between a set and its cardinal number affects the concept of transfinite number, and its usefulness.

Cantor starts section 6 (p.103 and further) immediately with the definition of some fundamental terms:

[part 1 of the definition]
"Aggregates with finite cardinal numbers are called "finite aggregates," all others we will call "transfinite aggregates" and their cardinal numbers "transfinite cardinal numbers."

[part 2 of the definition]
The first example of a transfinite aggregate is given by the totality of finite cardinal numbers v ; we call its cardinal number "Aleph-zero" and denote it by Z0 ; thus we define
(1) ZO={v}c"

The "totality of finite cardinal numbers" is a set that does not contain the finite cardinal numbers, but only their cardinal number, v, which Cantor calls aleph Zero. [Aleph zero is therefore neither a set of a number, but the name of such.]
In other words

i) {vc},[the set containing the cardinal number of all finite cardinal numbers]

will be different from

ii) {v} [which is an infinite, (?) Cantor does not say it yet, set of finite cardinal numbers.]

{v} must be understood as short for all finite numbers v1,v1...,v. Likewise
{v}c must be understood as short for: the cardinal number of {v1,v2,...v}.
For instance, the cardinal number of {1,2,3} is 3.

This has as a strange consequence that the cardinal number of v is v, and the cardinal number of {v} [{v}c] is 1, since {v} has only one member. 
This is of course only the case if we, just like Cantor is prone to do, try to save on the number of variables we use.
In this case, there is v as last [whatever that means in an infinite set] element of the list of cardinal numbers, and v as the cardinal number of all these cardinal numbers. The fact that they have the same numerical value does not make them one and the same object.

This is more than a simple stylistic remark, even if it started this way. We must not forget that for Cantor, as of statement (9), a set is equivalent to its cardinal member. We must therefore be prepared for many ambiguities in Cantor's arguments, wherever sets and cardinal numbers are concerned, which is in fact everywhere in his analysis.

Let us look again at the whole definition, even more closely this time.
Aleph Zero has only one member. Or more precisely, it is that member or number itself since it is not itself a set [even though it can be considered as one]! [we are now using Aleph Zero not as a name, but as the number it is the name of.]
Also, every member of v is a finite cardinal member, even though the list of all those finite numbers may itself be infinite. 
That brings us immediately to the conclusion that v cannot be an  cardinal number itself. Otherwise, Aleph Zero would be the cardinal number of a cardinal number,  which is 1. But in the expression "v1,v2,...,v" v is a indeed a finite cardinal number. The use of one and the same variable for a set and its cardinal number creates unnecessary contradictions.

To sum up:
v is a set whose members are the individual finite cardinal members. But wasn't that supposed to be {v}? Because otherwise, Aleph Zero would still be equal to 1. After all, the set {v} has only one member, as in {{1,2,3}}.

Let us take a concrete example again and let the "totality of finite cardinal numbers" be the set containing the following members 1,2,3,4; and let 4 be the highest cardinal number our bird brain can encompass. 4 is then as big as infinity can get as far as we are concerned. This way, the actual infinity for a bird brain will somehow symbolize our concept of actual infinity when looked at by a grinning djinn.

We conclude:
the cardinal number of v=4, or vc=4
{the cardinal number of v}={4}.[the set containing the cardinal number of v is equivalent to the set containing 4]. Or, {vc}={4}.

Which begs the question:
Aleph Zero={4} or
Aleph Zero=4?

Since Aleph Zero is a cardinal number, and not a set, it could not be possibly {4}. Cantor's definition (1) seems to be false. It should be
Aleph Zero= cardinal number of v. Or, Z0=vc.

This would be the case if we did not have 

"(9) M ~ Mc".

Or in this case, {v} ~ cardinal number of {v}; v taken as a shorthand for the different members not of v as a cardinal member, but of the set v of which v1,v1,,,v are members. 
The end result of this merry-go-round seems to be:

(9') Z0={4} AND Z0=4 AND Z0={vc) AND Z0=(v}c AND Z0=v AND Z0={v}.

The decision to accept (9) as valid has many disturbing consequences, to say the least.
The definition here makes it impossible to choose between Aleph Zero as being equal to 1 or to the cardinal number of all cardinal numbers. The fact that we know what Cantor wants is the only reason we can avoid the ambiguity, proposition (9) being unable to negate automatically other propositions.


Is Aleph zero a finite or a transfinite cardinal number? 
Cantor puts in a lot of efforts to prove that Aleph zero is a transfinite cardinal number. Before we look very closely at these proofs in the next entry 6.3, let us see what that entails for the list of finite cardinal numbers Aleph Zero is supposed to be a cardinal member of.

What is this "totality of final cardinal numbers"? It cannot be only the totality of the natural members, otherwise Cantor would have said so. But how could it be anything else?

Let us look at the so-called power set, the set of all subsets of the natural numbers 1,2,3. Its cardinal number, the number of subsets, is much larger that the number of its elements, even if we consider only the ordered subsets.

1 2 3
1 2
1   3
  2 3

Imagine that you are the dream of an anthropologist come true in that the only numbers you know are 1, 2 and 3. Once you have listed all the possibilities of the power set of {1,2,3), you realize that you have not enough numbers to count them! You have to invent new numbers! And that is exactly the situation Cantor imagines himself to be in. The totality of all cardinal numbers surpasses, according to him, the possibilities of the natural numbers. So, he invented Alephs and Omega's to be able to count sets beyond the realm of natural numbers.
The problem of course is the huge assumption he made that the natural numbers were not up to the job. His main argument, that an infinite set remains unchanged when we add a finite, however large, or even infinite [as in Z0+Z0=Z0] number of elements to it, was supposed to justify the creation of transfinite cardinal numbers, since it proved that a transfinite number is larger than any finite number. If my analysis is correct, his justification is certainly not sufficient to legitimize the introduction of transfinite numbers.

The problem with absence of proof, is that it is not proof of absence. There is no way that I could prove that there are no transfinite numbers. All I can do is follow Cantor very closely in his argumentation, and every time he claims the necessity of a transfinite number, I can try to show that natural numbers can do the same job.

For that I will follow the same double strategy I did until now:
1)The charitable approach where I try as objectively as I can to interpret Cantor's intentions in a way that makes his assertions meaningful.
2) A critical approach where I point at the mistakes and unwarranted assumptions in his argumentation.

One thing seems obvious to me until now, Cantor has certainly not succeeded in justifying the introduction of Aleph Zero. He will keep trying to use it, and it will be a challenge to meet each time he does it.

The Liar Paradox (and other beasties)
Cantor's Logic (6.3): The Transfinite Cardinal Numbers (Continued)

Proofs concerning Aleph Zero

"A. Every transfinite aggregate T has parts with the cardinal number ZO ."

Or, in plain English, 

A'. A transfinite set is made of infinite parts.

Let us look at the proof.

"Proof. If, by any rule, we have taken away a finite number of elements t,t1 t2,...,tv-1. there always remains the possibility of taking away a further element tv. The aggregate {tv}, where v denotes any finite cardinal number, is a part of T with the cardinal number ZO , because {tv}~{v}."

The first part simply says:

A".We can always take away one extra finite element out of an infinite set. 

Nothing world shaking so far. Even though it remains a very mysterious and unproven claim.
Let us assume for a moment that it is true. How could we ever get from the infinitely large to the infinitely small, or vice versa? There would be a discontinuity in the fabric of the universe that even Dedekind could not explain away.
I imagine that reals are somehow connected to natural whole numbers via their rational relatives. You can get from the whole to the part easily by having the denominator larger than the numerator, the inverse way being as simple.
Does then this affirmation, A", justify its counterpart, namely, that you can always add a finite number to an infinite negative number without ever reaching 0 or 1, much less positive infinity? That we humans cannot do that is pretty obvious, but is that necessarily so? 
[Please do not involve Kripke into this! It is already complicated enough!
Anyway, something to justify your headache to your partner. Unless of course you prefer to blame Canada.]

The second part of the proof:
"The aggregate {tv}, where v denotes any finite cardinal number, is a part of T with the cardinal number ZO , because {tv}~{v}."

This is really a Cantorian beauty. It says, as faithfully translated as I can manage:

The set which we will have taken away, and which had v members, is only a part (equivalent to cardinal number v) of T, which, as we know, has the cardinal number Z0.

Apparently, it is obvious to Cantor that we would still be left with Z0. But the whole point was to prove it, wasn't it?


"B. If S is a transfinite aggregate with the cardinal number ZO , and S1 is any transfinite part of S, then S1 = ZO"

The reader has to wonder whether B is a theorem that needs to be proven, or simply the definition of a technical term being "transfinite part of transfinite set".
That is not Cantor's position, he wants to prove that a transfinite part of a transfinite set is equal to aleph zero.
What does this position entail exactly?
If he can prove it, he will have shown that an infinite part is equal, has the same cardinal number aleph zero, as the set of which it is a part. Didn't we unveil a circularity in this approach somewhere along the way? 
First he needed to prove infinity by showing that some parts of an infinite set were themselves infinite. We took as example the case of even numbers put in a one to one correspondence with the infinite set of natural numbers.
If we think about this step for a minute, we will realize that we have never proven that the set of natural numbers was infinite. Mathematical Induction shows us that we can always add 1 to any n, and we take that as a valid indication that we are dealing with an infinite set. But such a conclusion is not warranted by Mathematical Induction which, if it proves anything, proves that we are always dealing with finite sets.
Infinity is something we superpose on the finite method that is Mathematical Induction.

Because Cantor was convinced that he had proven the equivalence of a subset with its mother set, he had a working definition of an infinite set, one that went back to Galileo before Bolzano worked it out in more details.
Also, Cantor assumes the truth of the statement that a transfinite set has aleph zero as cardinal member. It looks like a simple syllogism to me:

transfinite set= aleph zero
subset=transfinite set
subset=aleph zero.

What more is there to prove?

Cantor does not seem to want to make it easy on himself, he goes for a complicated strategy involving theorem G which we apparently did not treat with all the consideration it was due.
Allow me to use my own reformulation [in 5.5] to speed things a little bit up. Please feel free to check Cantor's own formulation (p.103).

Theorem G:
G. Every aggregate K = {k} of different finite number of elements can be brought into the form of a series 
K = (k1,k2,K3,...)
such that
k1 LT k2 LT k3,...

Why theorem G? I must admit my own perplexity and hope, just like you to find an answer in the text.
Allow me to reproduce at least a part of the proof [it is certainly not completely obvious where it ends, Cantor having forsaken any heading between this proof and the next one] to make it easier for me to quote specific parts.

"Proof. We have supposed that S ~ {v}. Choose a definite law of correspondence between these two aggregates, and, with this law, denote by sv that element of S which corresponds to the element v of
{v}, so that 
The part S1 of S consists of certain elements sk of S, and the totality of numbers k forms a transfinite part K of the aggregate {v}. By theorem G of [section 5] the aggregate K can be brought into the form of a series
kv LT kv+1;
consequently we have
S1={skv}. [The v is supposed to be a subscript of a subscript, something the text editor obstinately refuses.]
Hence follows that S1 ~ S, and therefore S1 = ZO."

Cantor creates a second subset to S, the subset K, which can be said to be an ordered set under theorem G.
The first subset S1, which Cantor is supposed to prove as having the cardinal number aleph zero just as its mother set S, is magically declared as being equivalent to the cardinal number v, which is the last element of K as well as the cardinal number of the mother set S. That is enough then to say of subset S1 that it is also equal to aleph zero!
[I think I know now where David Marr got his logic from! see the entry "Can somebody please tell me what codons are?" in my thread The Cerebellum .]

The Liar Paradox (and other beasties)
Reply to Eray Ozkural
Of course Mathematical Induction works! Why wouldn't it?

1 a1=x
2 a2=x
3 z?

Would you explain to me how an intelligent computer could arrive to the conclusion that z=x?
Allow me. You have to define, sometimes along the way, that z's are a's. It can be done in an explicit way, like a simple statement z=a, or it can be the result of a long chain of reasoning which eventually reaches the desired conclusion.
In fact, this conclusion might be something that you yourself have never thought of, but which the computer deduced out of its rules and data.
If you think that computers can arrive to this conclusion in a non-mechanical way, then you are probably the only one in your branch.
What that means is that the programmers have decided that mathematical induction was a valid rule, and turned in into a series of computer instructions. They put their beliefs into the computer as it were.

Computers may turn out one day to be as intelligent as humans. In fact, I see no reason why not as long as intelligence is defined as "intelligent behavior". They might even become "creative", if by creativity we mean the re-ordering of data in a random, but still meaningful way, meaningful being defined by other rules. No need to fear circularity, there are enough rules to go around.

Will that make computers "humanly" intelligent and creative? By the time we get there, the question will have become irrelevant, except for moralists and religious people, who will be either extinct, or the most powerful group in society. Pick your own scenario.

Until then I will just take note of your self-confidence and faith in the supremacy of mathematics.
Also, I could not hope but notice that all you do each time is state your positions, without ever justifying them. I am still waiting for an explanation how rejecting the axiom of infinity would solve all antinomies. I think you owe it to us. 
For the rest, I will let your convictions be your own.
P.s: I remember a cartoon I read in the paper many, many years ago. There was an anti-America group in a demonstration in Paris , holding up slogans like "Yankee go home!". An American tourist was watching the whole scene with a grin. Somebody asked him if that did not disturb him. He answered with a heavy southern accent: "why would it, I ain't a Yankee!"
Just you know, I am not a Platonist. 

The Liar Paradox (and other beasties)
Cantor's Logic (6.4): The Transfinite Cardinal Numbers (End)

More Proofs
Cantor starts a new, headless, section immediately after the one handled in 6.3 with this cryptic statement:

"From A and B the formula (4) results, if we have regard to [section] 2." (p.106)

This is what the formula says, with a being any other transfinite number besides aleph zero:

"(4) ZO LT a"

Section 2 concerns the concepts Greater and Less treated in Cantor's logic (1).
As for A and B, we might as well reproduce them again in their entirety:

"A. Every transfinite aggregate T has parts with the cardinal number ZO.
"B. If S is a transfinite aggregate with the cardinal number ZO , and S1 is any transfinite part of S, then S1ZO"

What is Cantor trying to prove, or at least show? How do A and B imply (4) by way of section 2?
A possible explanation would be that any transfinite number a other than aleph zero, will always have as one of its parts, aleph zero. Which means that the surplus will come above that common part, making transfinite set a greater than aleph zero.
But, before we get to any transfinite number different from aleph zero, we first have to determine the limits of the domain (here, more in the everyday sense, than in its function or set theoretic meaning) of aleph zero.
Cantor leads us from:

ZO+1=ZO; via
ZO+v=ZO; to
ZO+ZO=ZO; via
ZO.2=ZO  to
ZO.ZO=ZO and finally, via
ZO2=ZO to
All of this through small and, as far as I can see, valid logico-mathematical steps (P.106-107).

Proof of ZO.ZO=ZO

First, let me start with a very peculiar fact concerning Cantor's terminology.
When dealing with the addition of the sets M, N, Cantor calls the resulting set (M,N) "union aggregate" (Vereinigungsmenge). Even though he refers to M={m} [The first proposition of the article], he never uses expressions like 
M+N={m,n}, or (m,n).
He reserves this formulation for multiplication between two sets:

This can certainly not be considered as an essential point any way we look at it. Still, we have to be conscious of this idiosyncrasy when we encounter it. After all, there is no logical reason why (m,n) could not refer to the operation of addition, m+n. It seems like (M,N) refers to addition (Vereinigungsmenge) when whole sets are considered, and (M.N) to multiplication, "aggregate of bindings" (Verbindungsmenge), when sets or elements are concerned. Let us put all possible statements in a row:

(M,N) addition or union of two sets
(M.N) multiplication of two sets
(m,n) multiplication of two elements
unknown addition of two elements.

You wonder why Cantor did not keep the same logic for all these statements. Especially since it is bound to create ambiguities.

The first phase of the proof is to assert that

{(u, v)} ~ {L}. [I use L as in Lambda.]

with "{L}, {u}, and {v} are only different notations for the same aggregate of all finite numbers"

Note that this is nothing else but the reformulation of


The whole proof: [the numbering, using i's, of the sentences is mine, the numbering (Arabic numeral) of statements is in the original text.] p.107/108
i) Let us denote u + v by p ; 
ii) then p takes all the numerical values 2, 3, 4, . . ., 
iii) and there are in all p-1 elements (u, v) for which u + v = p, namely :
iv) (1,p-l), (2,p-2),..., (p-1, 1).
v) In this sequence imagine first the element (1, 1), for which p = 2, put, 
vi) then the two elements for which p = 3, 
vii) then the three elements for which p = 4, and so on. 
viii) Thus we get all the elements (u, v) in a simple series :
ix) (1,1);(1,2),(2,1);(1,3),(2,2),(3,1);(1,4),(2,3),...;
x) and here, as we easily see, the element (u, v) comes at the L'th place, where
(9)    L=u +  [(u+v-1)(u+v-2)]/2
xi) The variable L takes every numerical value 1, 2, 3,. ., once, xii) Consequently, by means of (9), a reciprocally univocal relation subsists between the aggregates {u} and {(u, v)}."

First two linguistic remarks:
The sentence (iii) should I think be understood as:
iii') and there are, in all, p-1 elements... [even in bold, the comma's before and after "in all" do not show properly]
All is not the quantifier of p-1 elements. The German text speaks of "im ganzen p-1 Elemente..." and not of "in den ganzen p-1 Elementen ..."

Sentence (v) would have been more elegant by leaving  the "put" away. It does not add anything new to the meaning of "imagine", and sounds rather strange in the English sentence. 

And now the same stylistic remarks as before: Cantor uses u and v both as sets and as elements, which creates the necessary ambiguities already mentioned.
Here (u, v), which should be used solely to denote multiplication, is also used in an addition operation.
We will not hold it against the author, but keep vigilant as to which meaning he is using each time.
Beware that he is using the same formula (u,v) in the sense of multiplication (aggregate of bindings or Verbindingsmenge) in the statement he wants ultimately to prove ({(u, v)} ~ {L}), and in the sense of addition (union-aggregate or Vereinigungsmenge) in the body of the proof itself.

Just as by Theorem G [see 6.3], we wonder why Cantor needs such a formula (9) and all that it entails. He gives no reason and jumps immediately in the sea, expecting us to follow or drown. Where is he swimming to?

Well, it does make more sense if we take the ambiguity away, and interpret (u,v) as meaning (u.v)

We have to start with

xx) {(u.v)} ~ {L}; then go to 
xxi) {(u.v)} ~ {u}; since u~v~L; and that is is what Cantor need to prove.

Let us see how he fares (or swims).

We will use the structure of his own argument as presented above.

We will take as maximum values u=3 and v=3

i) 3+3=6
ii) p=2,3,4,5,6
iii) The number of possible values of p without repetition = p-1 = 5.
iv) The following pairs are possible, with the corresponding p (and L) values: 
u v p L
1,1 2 1
1,2 3 2
1,3 4 4
2,1 3 3
2,2 4 5
2,3 5 8
3,1 4 6
3,2 5 9
3,3 6 13
v-ix) (1,1);(1,2),(2,1);(1,3),(2,2),(3,1);(2,3),(3,2);(3,3)

Now we can see what conclusions we can draw from this argumentation.
First, u, v and L are supposed to be infinite. They represent the same set of all finite cardinal numbers. 
Unfortunately for Cantor, the formula (9) does not guarantee this equivalence. Our simple sample of values from 1 to 3 shows the number 7 to be missing. It is not to be expected that it could appear later on in the series. L is therefore not the set of all cardinal numbers, and as such not equivalent to u and v.

Anyway, let us assume that I have made a mistake somewhere and that L does contain all finite cardinal numbers.

(9) L=u + {[(u+v-1)(u+v-2)]}/2; can be understood as 

(90) L= (u+v),
where v is a complex formula involving both u and v and operations of addition, subtraction, division and multiplication. We could consider only the multiplication as essential for this argumentation. Which means that we would interpret (9) this time as

(91) L= u+(u.v), 

with the v and u between the parentheses being themselves formulas.

This is quite an arbitrary and very opportunistic interpretation, still it is quite legitimate as the following example shows:

for             L= 1 + {[(1+2-1)(1+2-2)] / 2}.
we would have   L= u + {(u1)(v1)}
with            u1= [(1+2-1)]/2 
and             v1= [(1+2-2)]/2

This way, we would have proved that "a reciprocally univocal relation subsists between the aggregates {u} and {(u, v)}".
We would not even need L to be equivalent to u and v, so even if it appears that some cardinal numbers are missing from the list, that would be totally irrelevant.

The problem is that this result is still not what we are looking for, or what Cantor had set out to prove, which was:

(u.v) ~ u.

Let us turn L into u, we would then have

92) u = u + {(u1)(v1)}; or even better
93) u = u + (u.v).

We are now proving more than we intended to since all we needed was u=(u.v).

Another problem, not so easily ignored, is that this conclusion seems to be applicable to finite sets as well, as shown by the concrete examples above.

So, even if we ignore the fact that L does not contain all cardinal numbers, we are either proving too much, or not enough, the formula (9) being valid for finite and infinite sets, which makes it unusable for proving a property specific to infinite sets.

Still, in the spirit of charity, we could say that Cantor has proven his point once we massage away some imperfections in his presentation.
- we do not need L to be equivalent to u and v;
- u=u+(u.v) does prove that we can add u.v and more than u.v to the original u, and still keep u. [u dig?]

So, we could maybe say that Cantor has proven his point, even if in a clumsy way.
Of course, all of this depends on the validity on what was said before. So, it still does not sound too good for Cantor's conception of the transfinite.
Furthermore, we have to contend with the fact that the formula is applicable to all sets, finite and infinite. So, what does it prove, really? Any variable can be reinterpreted and translated into a formula applied to itself, so long as it is not a single operation like multiplication. And that is exactly what Cantor needed.


Even more proofs:

Let us first recapitulate the theorems A and B since we will need them for this proof;
"A. Every transfinite aggregate T has parts with the cardinal number ZO."
"B. If S is a transfinite aggregate with the cardinal number ZO , and S1 is any transfinite part of S, then
S1c = ZO." [cardinal number S1 = aleph zero]

Please remember that I showed in 6.3 that Cantor's treatment of both these theorems was very unsatisfactory, to put it mildly.
Nonetheless, I will attempt first to consider them as proven.

"D. Every transfinite aggregate T is such that it has parts T1 which are equivalent to it."

I will reformulate the proof to make it less ambiguous. Please refer to p.108 for the original formulation.

i)    By theorem A of this paragraph there is a part S = {tv } of T such that Sc=Aleph Zero.
ii)   Let T = (S, U), so that U is composed of those elements of T which are different from the elements t1,t2,...,tv
iii)  Let us put S1 = {tv+1 }, 
iv)   T1 =(S1, U); then
v)    T1 is a part of T, and, in fact, that part which arises out of T if we leave out the single element t1
vi)   Since S ~ S1, by theorem B of this paragraph, and
vii)  U~U, we have, by {section] 1,
viii) T ~ T1.

Translated in symbols the argument looks like this:
0)    Sc=Z0
i)    S ={ t1,t2,...,tv}
ii)   T ={(t1,t2,...,tv), U}
iii)  S1={ t1,t2,...,tv , tv+1}
iv)   T1={(t1,t2,...,tv), tv+1), U}
v)    T1=T-t1 
vi)   { t1,t2,...,tv} ~ { t1,t2,...,tv , tv+1}
vii)  U ~ U
viii) T ~ T1

Still assuming the validity of theorems A and B, I would like to make the following remarks.
- Theorem A, B and D look suspiciously like different formulations of the same main idea. But then it would cost as much energy to try to prove it as it would to analyze each theorem separately.

- Statement (iv) does not seem correct, tv+1 must come from somewhere. In fact, it can only come from U. We should then have U-1 here. Assuming that it does not matter is circular, we would be assuming that which we want to prove. Unless of course we appeal to previous theorems, in which case adding anything to the first part of the formula would still leave U unchanged. Still, that would make the whole process very suspect.

- Statement (v) is very interesting. What are we subtracting exactly from T? Here it would appear to be a unity, t1. Which means that adding t1 to both sides of the equation would give us

T1+t1 = T therefore, assuming t1=1,
(iv) {(t1,t2,...,tv), tv+1), U} +1 = T
(ii) {(t1,t2,...,tv),        U}    = T

Could we deduce the value of U?  It could have a negative value equal to tv+1 +1. But then we would get in trouble with the final value of the whole formula. And since +1 is by definition not zero, the same being said of tv+1, we are really in trouble. All those formula's, when put together, do not make any sense. Whatever the rules governing the creation of aleph zero might be, they do not seem to be mathematical or logical.
We really need the validity of the previous theorems A and B to resolve the problem. But when we do that we put logic over board, and then everything seems possible.

- The proof is unnecessarily complicated. The obvious shortcut is:

ii)   T  = (S, U)
iv)  T1= (S1,U)
vi)  S = S1
viii) T = T1

That is in fact all the conclusion is stating, which leaves the role of t1 and tv+1 in the mist. They might as well be considered as simply absent. After all, S and S1, just like T1, are part of T, and as such could be considered as identical to it.
The three statements that constitute the core of the argumentation (ii, iv, viii) were all brought in by definition or supposedly proven theorems.

In summary: on the condition that the previous theorems be considered as valid, Cantor's proof of D could be said to be valid. Just like a tautology is always considered as valid. The proof of D, just like so many Cantorian proofs, is a restating of the obvious, or what Cantor considers as such.
I am afraid that under a critical perspective I would have to say that Cantor has not made any progress in his justification of transfinite numbers.

The Liar Paradox (and other beasties)
Cantor's Logic (7) The Ordinal Numbers 

[p.110 ff]
Ordered sets have ordinal types. The type of a set M is denoted by ordinal M, with a single bar, to distinguish it from the cardinal numbers who have two bars. I will use Mo whenever possible. To avoid confusion with M0, I will sometimes use the expression ordinal M.

"By [ordinal type] we understand the general concept which results from M if we only abstract from the nature of the elements m, and retain the order of precedence among them." (p.112)

This tells us here that "ordinal type" is not a number, but some property of sets (and numbers).
Still, "ordinal type" is more than a simple predicate, it is also used to denote a set, the members of which are ordered according to such a type:

"Thus the ordinal type M is itself an ordered aggregate whose elements are units which have the same order of precedence amongst one another as the corresponding elements of M, from which they are derived by abstraction."
[By the way, the German text speaks of "Einsen", ones, and not of "Einheiten", units. I agree with the translation though, how could there ever be an order of precedence between "ones"?]
Ordinal M is therefore an abstract duplicate of a set M, where only the order of precedence is taken into consideration. The other, concrete or abstract properties of the elements of M are therefore considered as irrelevant.
Except of course, their number, or more precisely their cardinal number. 

That brings us to a very peculiar relationship between both concepts. Cantor states clearly:

"If, with an ordinal type Mo we also abstract from the order of precedence of the elements, we get [...] the cardinal number Mc of the ordered aggregate M, which is, at the same time, the cardinal number of the ordinal type Mo." p.113.

In short, order, or ordinality, does not change cardinality, even though it can hide it temporarily. But when we abstract order away, we find cardinality again.  Also, "ordered aggregates of equal types always have the same power or cardinal number".

This means that we can speak of ordinal numbers, they are the ones that change into cardinal numbers when "we also abstract from the order of precedence". 
This time, Cantor does not try to convince us that we are dealing with a new species of numbers, at least, not as long we are considering only finite sets:

"Thus the finite simple ordinal types are subject to the same laws as the finite cardinal numbers, and it is allowable to use the same signs I, 2, 3, . . ., v,... for them, although they are conceptually different from the cardinal numbers." 

Here is a terminological rule: 
"If a is an ordinal type, we understand by
(5) ao [single bar]
its corresponding cardinal number."
Not its ordinal number which is, as stated, simply a. Not really obvious, and therefore something to remember.
Also, all finite ordered sets with the same cardinal number have the same ordinal type. That's a whole lotta types out there. As many as there are finite cardinal numbers. Or more, or less, a lot!

But when it comes to transfinite numbers, then we find out that one and the same (cardinal) number of elements can be ordered very differently. 
"for to one and the same cardinal number belong innumerably many different types of simply ordered aggregates, which, in their totality, constitute a particular "class of types" (Typenclasse)."

This is I think where it gets a little bit dicey, even if the language still sounds quite ordinary.

"Every one of these classes of types is, therefore, determined by the transfinite cardinal number a which is common to all the types belonging to the class."

What could that possibly mean? We have, until now, 
- sets with all the same cardinal number;
- each set can have its own ordering principle (s);
- to each ordering corresponds an ordinal type;
- finite simply ordered sets all belong to the same type;
- tranfinite types can themselves be put in a set: a class of types.
- to each class of types corresponds a transfinite cardinal number.

There, I said it. Still shaking all over, because I still have no idea what it all really means.
And I was right to worry. Even before we had time to digest the news, Cantor tells us that these transfinite ordinal types belong to a transfinite cardinal number a, and are therefore all members of the "class of types [a]".
He adds immediately that what we are interested in is in fact "the class of types [ZO ] which embraces all the types with the least transfinite cardinal number ZO."

[I am not so sure about that. I mean, I would not mind getting to know the first group a little better. Do some binding, bonding, bondage, whatever. I saw some cute sets over there! And they all had names starting with a v, like Veronica, Valery, virgins. Oops, my bad. Still, those transsexual types, I mean transfinite, are all like Arnold, Asphodel  and Armed to the teeth!
George (whispering): that's it! You're getting us banned!
me (not whispering): Why? Baby what'd I say? ...? George? What...
George (still whispering, sounding scared): I think I'll go back to Cognitive Sciences. Maybe they'll let me stay even if they sack you! I missed you, you know.(Shakes his head).
Pfft! Sound of George disappearing.
me (shouting and shaking my fist at the emptiness between my ears): you... you... you homunculus!
me (whispering): I miss you too.]

That my own fears were not groundless is shown by the fact that the relation between [a] and [alpha zero] is at the moment still unclear. The text, in the original text even more than in the translation, makes it sound like [Z0] is a part of [a] types. Let's hope this issue will be cleared up soon. One think for sure, they all concern transfinite cardinals and ordinals.
The fog certainly does not dissipate with explanations like this:

"From the cardinal number which determines the class of types [a] we have to distinguish that cardinal number a' which for its part is determined by the class of types [a]. The latter is the cardinal number which [section 1] the class [a] has, in so far as it  represents a well-defined aggregate whose elements are all the types a with the cardinal number a. We will see that a' is different from a, and indeed always greater than a."

let's see if we can make sense of this mess. [There really should be a law against writing this kind of sentences, not to say publishing them. But seen as they are probably sponsored by the International Firearms Association there is very little hope for that.]

Cantor seems to be saying two things:
- There are sets whose members have all the cardinal number a;
- these sets have themselves a cardinal number, which depends not on the cardinal... forget the last part. It will only make things more complicated. The sets also have a cardinal number. That is all we need to know for now.
Either the members or the sets are of the type [a], respectively [a']. Let us also leave it at that for the time being.

Section 7 is obviously no more than an introduction to the subject of ordinal numbers and its terminology. Better not to try and understand everything at once.

[And please? tell George it is all right to come back?]

The Liar Paradox (and other beasties)
Gödel's proof in "On formally Undecidable Propositions of Principia Mathematica and Related Systems"
The proof can be summarized as follows:
All statements of a logical system can be translated into natural numbers.
"the formula v is not provable" is such a statement and can therefore be translated into a natural number.
Any logical system can be considered as a series of natural numbers whose members are created by the translation procedure.
If "x is not provable" is a member of this series, we are confronted with a contradiction. Either the statement is true, and therefore it is not provable. or it is not true, and it is provable while it claims to be unprovable.
As Gödel himself admits:
"The analogy between this result and Richard's antinomy leaps to the eye; there is also a close relationship with the "liar" antinomy..." (p.40) 
I find the footnote belonging to this sentence, footnote 14, particularly interesting: "Every epistemological antinomy can likewise be used for a similar undecidability proof." 

I would like to refer the reader to the entries at the beginning of this thread. I think that my remarks about reference and self-reference are certainly applicable to Gödel's proof.

I find the following affirmation not really convincing:
"In spite of appearances, there is nothing circular about such a proposition, since it begins by asserting the unprovability of a wholly determinate formula (namely the q-th in the alphabetical arrangement with a definite substitution), and only subsequently (and in some way by accident) does it emerge that this formula is precisely that by which the proposition was itself expressed." (p.41 footnote 15)

Still, I will not attempt to prove the circularity of the argumentation, but nonetheless unveil its fundamental flaws.

The whole proof relies on the principle that it is possible to express, mechanically, whether a proposition is provable or not. More than that, it is supposed to be possible for a system to express that one of its own propositions is undecidable.
That is a fact which is very often overlooked. People usually concentrate on the idea that a system cannot decide of its own truth or falsity, forgetting that: "the proposition which is undecidable in the system PM yet turns out to be decided by metamathematical considerations."(p.41)

Since the whole proof depends on the legitimacy of the proposition "is (not) provable", we must ask ourselves if such a proposition can indeed be part of a logical system, and therefore, whether the Gödelian translation schema is valid.

Let us take a simple logical rule which will be considered as the paradigm of all possible logical propositions. That should make it easy for critics to attack my approach.

(1) "If a=b and b=c then a=c"

Let us try and find a mechanical procedure to decide whether such a proposition is provable. 
First, let us assume that "provable" means that we can assign to the proposition one, and only one of the two predicates True or False (in the logical system considered). Otherwise it should get the predicate "undecidable".

We see immediately our dilemma. We would need a rule that tells us whether (1) is true or false. But there is no such rule except (1) itself. The proposition expresses and embodies the rule. Nothing we could say would be expressible in words different from (1).

(2) The proposition "If a=b and b=c then a=c" is true.

will always either be true, or it would denote a mental illness or dishonesty from the part of the utterer of the expression, as in 

(3) The proposition "If a=b and b=c then a=c" is false.

What about the following propositions? 

(4) The proposition "If a=b and b#c then a=c" is true 

(5) The proposition "If a=b and b#c then a=c" is true" is true.

(6) The proposition "If a=b and b#c then a=c" is true" is false.

(7) The proposition "The proposition "If a=b and b#c then a=c" is true" is true" is true.

(8) The proposition "The proposition "If a=b and b#c then a=c" is true" is false" is false.

We could go on indefinitely.

If we can assume that all propositions in PM will be of the kind of (1) we will realize that none of the sentences 2-8 can itself be part of the system itself. They are either utterances of the logician, and therefore not a part of the system he is building. Or they are indeed a part of the system itself, and as metamathematical judgments, will always be true unless the logician has made a mistake somewhere.
A proposition like " x is undecidable" will follow the same rules: either a true utterance, or a true metamathematical proposition which has been made part of the system, after having been recognized as a true utterance.

Where does that leave Gödel's proof?

The Liar Paradox (and other beasties)
What is Diagonalization good for?

This method, invented, or at least perfected by Cantor, has been considered as a jewel of mathematical and logical argumentation for more than hundred years. There have been countless books on the subject, and any student not capable to grasp its meaning would be considered unfit for a Mathematics or Logic degree.
I have certainly not the pretension of criticizing the validity of the procedure as such. If there are flaws somewhere I lack the expertise to find them.
What I tried to show in my analysis of Gödel's proof was that its application is not a technical matter. The Logician and mathematician both have to apply it consciously and not in an automated way. By that I mean that the choice of the logical and mathematical statements it can be applied to is not something the method itself can decide.
That will sound very strange in the ears of many. After all, isn't that which Gödel's proof is all about? Deciding whether a logical statement is "logical", in the sense of provable? Isn't that what Cantor's proof is about, deciding whether reals are countable?
I hope that I have shown that both cases are somewhat different from each other. Cantor can justly claim that he has shown that reals cannot be counted the way other sets of natural numbers can be. His results are, as far as I can see, undeniable. The theoretical, philosophical and metaphysical considerations he attaches to his conclusions are his own. They are the result of other processes than the technical properties of the diagonalization method. 
Also, his definition of countable was very precise (one to one correspondence with natural numbers, and that in one go).

Gödel on the other hand is trying to apply the method illegally. He gives the right to the diagonalization method to chooses its own objects and field of application. That he cannot do. His conception of what is provable, or what is not, is not a technical matter that can be solved by any method. Ultimately, the Logician has to decide of the validity of logical statements. The illusion that a mechanical procedure exists that would render the Logician superfluous is only that. An illusion.
The Logician's behavior can of course itself be an object of study and automatization. In the previous entry

(2) The proposition "If a=b and b=c then a=c" is true.

could become a computer rule, and as such play a role in the mechanical  processing of logical arguments. But even then, (2) can never become an object of the diagonalization procedure, for the simple reason that it is not a logical statement, whatever Gödel would like us to believe.
Instead of computer programs, think of what a sociologist or psychologist would make of (2). I certainly agree with Frege that there must be a distinction between sociological and psychological truths on one side, and logical truths on the other. This fundamental distinction I see expressed in the difference between (2) and

(1) "If a=b and b=c then a=c"

This last statement is a logical truth, while (2) is (only) its verbal confirmation (which can be the legitimate object of psychological or sociological study). (1) does not need (2) to be true, the same cannot be said of (2).

What about Tarski's tables?
[C stands for "A OR B"]


How does this table compare to (1)?

(10) "If a=t and b=t then a or b = t"
(11) "If a=t and b=f then a or b = t"
(12) "If a=f and b=t then a or b = t"
(13) "If a=f and b=f then a or b = f"

What would the meaning be of (20)?

(20) "If a=t and b=t then a or b =t" is true

It sounds like the repetition of C, and does not seem to add anything to the table or its translations. 

Let us take a more sophisticated example:

i) if a=undecidable  and b=undecidable then c=undecidable

could we then say

ii) "if a=undecidable  and b=undecidable then c=undecidable" is true


iii) "if a=undecidable  and b=undecidable then c=undecidable" is undecidable

Maybe we would then realize that "undecidable" can be understood as the name of a variable, just like in (i), or as a specific meaning.
All these different  statements make us realize the difference between "undecidable" as a property comparable to "true", and its use as a variable equivalent to any other variable for which it can be substituted, and inversely.

"True" and "undecidable" are not variables of the system. They are primitives, and as such, make the system, and the diagonalization method possible. Just like "number". You can ask of a method to decide whether an object belongs to one of the known number sets, but you certainly cannot ask it to decide if an unknown object is a number. The same way, you cannot ask of a method if a statement is true unless the rules of the system leaves no ambiguity as to what is true and what is not. Which, I will concede that to Gödel, is impossible. But even if that was possible, no method could ever point at a statement and label it as "undecidable" unless that term has also been defined without ambiguity. Otherwise the method will be unable to decide whether a statement is "undecidable".

I admit that this whole argumentation sounds like playing the Gödelian devil's advocate. But that would be a very superficial conclusion.
When Gödel's method labels a statement as "undecidable", said method is giving each statement a very specific place in its system which comprises three categories:
- true statements,
- false statements,
- undecidable statements.

What is happening here is that the method is unable to decide where to put some statements. That says nothing about the statements, and all about the method which then hereby proves itself to be unreliable. A computer program would probably just get stuck in an endless loop. A clear indication that we are facing a version of the Halting Problem. Adding a fourth category would only postpone the inevitable.

The point is that I completely agree with Gödel's conception of the limits of any logic system. I just do not think that you can prove this point in a mechanical way.

The Liar Paradox (and other beasties)
Cantor's Logic (8) Addition and Multiplication of Ordinal Types

In section 7 Cantor gave the example of two types of sets, one in which types are similar to themselves in only one way, and in the other in an infinity of ways. It remains a mystery until now what this distinction is and what its consequences are.
"Not only all finite types, but the types of transfinite "well-ordered aggregates," which will occupy us later and which we call transfinite "ordinal numbers," are such that they allow only a single imaging on themselves. On the other hand, the type n is similar to itself in an infinity of ways." p.115.

[a -< b means, a precedes b] 

The two examples are:
"By w [omega] we understand the type of a well-ordered aggregate
(e1,e2,...,ev,... ),
in which
ev -< ev+1
and where v represents all finite cardinal numbers in turn." 

The second example being:
"On the other hand, let us take an ordered aggregate of the form
where v represents all positive and negative finite integers, including 0, and where likewise
ev -< ev+1."

We are dealing therefore with cardinal numbers on one hand, and finite signed integers, including 0, on the other.
In the first case, that of cardinal numbers, two sets of the same type will be similar if their elements, and their order, correspond to each other: lowest element in one, to lowest element in the other, and so on.

The second case concerns "normal" numbers, that is, not transfinite.

The first type, omega, is incongruous in that the law of commutativity does not seem to work, while other laws, like associativity and distributivity, do work.
We have therefore:
w+1 # 1+ w; 

How is that possible?
Remember, we are dealing with

E     = (e1,e2,...,ev,... ), "but, says Cantor, the aggregate, [f being a new element]
(f,E) = (e1,e2,. ,ev, ...)                is similar to the aggregate E, and consequently
1+w   = w"

He continues with what he considers as another evidence:
"On the contrary, the aggregates E and (E, f) are not similar, because the first has no term which is highest in rank, but the second has the highest term f. Thus w+1 is different from 1+w."
You could torture your brains as long as you can stand it, a couple of days did it for me, you won't find any rational explanation to this distinction if you do not assume that transfinite numbers remain unchanged when a quantity, however large, is added to them.

Still, how is "(f,E) = E" possible?
Assume f=1, then all we need to do is start with 1 and then go on with the infinite elements of E. In other words, E would not even notice the new passenger f trailing behind.
But try now to add 1 to E, which is infinite. That is certainly a different matter.
At least, that is what Cantor wants us to believe.
But wasn't E an actual infinity? In this case, why should E+1 be any different from 1+E?
I will concede that adding infinity to 1 seems more feasible that adding 1 to infinity. In the first case, the extra element is immediately integrated in the new set, while in the second it is delayed indefinitely, or more precisely, infinitely. 
But that is really splitting hairs. Both operations are hardly conceivable, and it seems that Cantor only uses actual infinity when it is convenient to him.

Then we think of the fact that we are dealing with ordered sets, which might make the distinction between w+1 and 1+w much more understandable. After all, it is a different order in both cases, isn't it?

Let us see. We have 1 and then we add {1,2,...} to it. Where shall we put this new 1? Um, let me think... At the beginning of course!
We would then have {1,1,2,...}. But that would make E # (f, E)! The only way to keep the identity would be if all the elements of E would be so kind as to shift one place to the right, the 1 becoming 2 becoming 3..., and leave a vacant seat for the new guy to settle in. How could that happen? The new 1 would only be able to take its place once all the elements of E will have finished their shimmy. Which would I am afraid take an infinitely long time.

Nevermind. Let us look at E+f. With f still equal to 1, I wonder what the problem really is. Why does it matter if we put it at the end or at the beginning? 
Still, let us look more closely at this second case. Let us assume that because E is an actual infinity, we are able somehow to append the new 1 to E. How would we do that? We cannot leave the 1 just dangling at the end of infinity, we have to make it part of a number like v+1. But, just as in the previous case, what to do with the already existing v+1? The fact that it is somewhere yonder in the infinite won't make it move before its successor has moved, and its successor'successor has moved, and so on. Unless we give up on Infinity altogether.
Both cases are in fact identical, and both seem as improbable and problematic as the other.

That is the thing with Infinity, it does not move so easily. Talking about aleph's or omega's does not change this hard fact.
Once again, Cantorian proofs would appear to be nothing else but the obscure reformulation of his own prejudices.

[I will not treat of the multiplication of ordinals, seen as the argumentation is exactly the same as with addition, commutativity being also seen as invalid.

In fact I propose to end the analysis of the first article here. The remaining sections either do not solely concern transfinite numbers, or do not contain anything new to justify the same kind of close analysis as practiced until now. 
I will therefore start with the second article in the following entry, but reserve the right to come back to the first one if the circumstances demand it.]

The Liar Paradox (and other beasties)
Cantor's Logic (12): Cantor's So-Called Proofs.

[I skipped section 9-11, but left the numbering intact to keep the direct link with the articles.]

The second and last article of the "Contributions to the Founding of the Theory of transfinite Numbers" starts with a very ambitious statement which we are not yet capable of appreciating fully:
""ordinal numbers" form the natural material for an exact definition of the higher transfinite cardinal numbers or powers". That will have to wait until Cantor gives us more clarity on his intentions. For now I propose to look at this section where he specifies what the properties of ordinal sets are. He speaks each time of "proof". What does the word mean when used by Cantor? I will analyze all these so called proofs, going from A to E, and try to formulate an answer to my question: what are Cantorian proofs in the Contributions?
First, some definitions:
There are simply ordered sets:

"We call an aggregate M "simply ordered " if a definite "order of precedence" (Rangordnung) rules over its elements m, so that, of every two elements m1 and m2 one takes the "lower" and the other the "higher" rank, and so that, if of three elements m1, m2, and m3, m1say, is of lower rank than m2, and m2 is of lower rank than m3, then m1 is of lower rank than m3." (p.110)

And then there are "well-ordered sets":
"We call a simply ordered aggregate F [...] " well-ordered " if its elements f ascend in a definite succession from a lowest f1 in such a way that :
I. There is in F an element f1 which is lowest in rank.
II. If F' is any part of F and if F has one or many elements of higher rank than all elements of F', then there is an element f' of F which follows immediately after the totality F', so that no elements in rank between f' and F' occur in F."
Let me draw your attention to the similarity of this last part with the famous Dedekind's cut. Now that I have done so, you may direct your attention again to what follows. Just remember it as Cantor's cut.

Proof of A

"A. Every part F1 of a well-ordered aggregate F has a lowest element.
Proof. If the lowest element f1 of F belongs to F1 , then it is also the lowest element of F1
In the other case, let F' be the totality of all elements of F [The translation says mistakenly F'] which have a lower rank than all elements F1, then, for this reason, no element of F lies between F' and F1. Thus, if f' follows (...) next after F', then it belongs necessarily to F1 [The translation says mistakenly F] and here takes the lowest rank." (p.139)

[Two fundamental mistakes in the translation of a few lines. Unless they are of course, which is more probable, simply overlooked typos.]

Let us dissect the argumentation as usual.
i) If the lowest element f1 of F belongs to F1 , then it is also the lowest element of F1
ii) In the other case, 
iii) let F' be the totality of all elements of F  which have a lower rank than all elements F1
iv) then, for this reason, no element of F lies between F' and F1
v) Thus, if f' follows (...) next after F', then it belongs necessarily to F1 and here takes the lowest rank.

Let us now take a concrete example.
0)   F ={1,2,3,4,5,6}

i)   F1={1,2,3} lowest element of F belongs also to F1. Done with this case.
ii)  F1={3,4,5,6}; second case: lowest element of F does not belong to F1
iii) F'= {1,2}; lower than all elements of F1.
iv)  no element of F lies between F' and F1. [nothing between 2 and 3].
v)   f'= {3}; comes right after F'; belongs to F1; lowest in rank in F1.

This is really a beautiful proof, or should i say elegant? How does Cantor come up with such proofs! Especially when the thing to prove sounds so obvious: 
a) "every part of a well-ordered set has a lowest element"

Compare it with

b) "every part of a set whose elements go from lowest to highest has a lowest element."

What does (b) mean exactly? 

c) However you cut such a well-ordered set, the subset will go from lowest to highest. 

Duh! What else would you expect?

Look at our example:
Imagine now that you are taking any part out of this set. Can you imagine that part without a lowest element?
So, what is the difference between the first case, the obvious one, described in (i), and the second one? Why does this second case need a proof? 

Maybe we should ask ourselves why Cantor is introducing an extra element in the argumentation that somehow makes it more complicated? Maybe it is because he then can solve the problem he has created willfully? But does he really?

In this case it is the subset F' which comes unexpectedly in the picture. What is exactly its role in the argumentation? Could Cantor have reached the same conclusions without this extra subset?
Remember, he wants to prove 

"A. Every part F1 of a well-ordered aggregate F has a lowest element."

Let us get rid of the first, obvious case, and of the extra subset F'. We are then left with very little!

0)   F ={1,2,3,4,5,6}
ii')  F1={3,4,5,6}; second case: lowest element of F does not belong to F1

how do we prove that F1 has a lowest element without using an extra subset? 

F' is used to get all elements lower than F1. But isn't that exactly what we are supposed to prove?
How do we get F' itself? How do we know that it will contain only elements that are lower than F1?

If we can know that, then why could we not know that F1 has a lowest element? Without that knowledge, we would after all not be able to pick the right elements for F', "right" meaning "lower than all the elements of F1".

Assuming the presence of F' does not therefore solve the problem, even if it falsely gives that impression. F' only makes sense if we had already solved the issue of the lowest element of F1. And that is exactly what Cantor had set out to prove!

Can we do better? Can we prove A? Not really. In fact, I think that this problem, in this form at least, is unsolvable. How are we to determine the lowest element of a set without direct knowledge of the set? And If we happen to get that direct knowledge, why would we need to prove it? If we could, it would not be direct knowledge anymore.
Cantor thought that by introducing an intermediate step he could avoid the problem, but all all he did was postpone it. No matter how many step we would insert, we will always be confronted with the necessity of knowing first hand what a set looks like, what are its lowest and its highest elements. That is empirical knowledge that no proof can provide.

Proof of B

"B. If a simply ordered aggregate F is such that both F and every one of its parts have a lowest element, then F is a well-ordered aggregate.
Proof. Since F has a lowest element, the condition I is satisfied. Let F' be a part of F such that there are in F one or more elements
which follow F' ; let F1 be the totality of all these elements and f' the lowest element of F1, then obviously f' is the element of F which follows next to F'. Consequently, the condition II is also satisfied, and therefore F is a well-ordered aggregate." (p.139-140)

i)   F has a lowest element, the condition I is satisfied. 
ii)  Let F' be a part of F such that there are in F one or more elements which follow F' ; 
iii) let F1 be the totality of all these elements 
iv)  and f' the lowest element of F1
v)   then obviously f' is the element of F which follows next to F'. 
vi)  Consequently, the condition II is also satisfied, 
vii) and therefore F is a well-ordered aggregate.

ii)  F'= {1,2,3}; F= {1,2,3,4,5}
iii) F1= {4,5}
iv)  f'= {4}
v)   f' follows F': {4} follows {1,2,3}

I wonder. Are we here dealing with a proof, or with the construction of a well-ordered set? The conclusion is inescapable for the simple reason that all the preceding steps are instructions, step by step, on how to construct such a set: 

ii)  let ...
iii) let ...
iv)  let ...
v)   then obviously ...
vi)  consequently ...
vii) and therefore ...

It could not be otherwise since B is in fact nothing more but a description of what is allegedly supposed to be proven:
- F is a simply ordered set: condition I is met;
- both F, by (i), and every one of its parts, by (iv), have a lowest element: condition II is met thanks to f'.


Proof of C

"C. Every part F' of a well-ordered aggregate F is also a well-ordered aggregate.
Proof. By theorem A, the aggregate F' as well as every part F" of F' (since it is also a part of F) has a lowest element ; thus by theorem B, the aggregate F' is well-ordered." (p.140)

Since theorem A was shown as unsolvable without direct knowledge, and theorem B was more a set of guidelines how to construct a well-ordered set, we can say that the proof of C does not really amount to much.

[see Proof of E for a more complete discussion of proof by construction.]


Proof of D
"D. Every aggregate G which is similar to a well-ordered aggregate F is also a well-ordered aggregate.
Proof. If M is an aggregate which has a lowest element, then, as immediately follows from the concept of similarity [...], every aggregate N similar to it has a lowest element. 
Since, now, we are to have G similar to F, and F has, since it is a well-ordered aggregate, a lowest element, the same holds of G. 
Thus also every part G' of G has a lowest element ; for in an imaging of G on F, to the aggregate G' corresponds a part F' of F as image, so that 
G'similar to F'. 
But, by theorem A, F' has a lowest element, and therefore also G' has. Thus, both G and every part [G', omitted in the translation] of G have lowest elements. By theorem B, consequently, G is a well-ordered aggregate." (p.140)

We are seeing here the same approach: an obvious statement which amounts to a definition is turned into a theorem to be proven.
As soon as I read D I thought by myself: you could never prove that. The statement is in itself self-sufficient. Once you know what a well-ordered set is, you know what a set similar to it is. The so-called proof is an obvious illustration of this principle. 
To know what I mean I could ask you to look at both

1) A
2) A

Now try to prove to me that the first A is similar to the second A. 


Proof of E

"E. If in a well-ordered aggregate G, in place of its elements g well-ordered aggregates are substituted in such a way that, if Fg and Fg' are the well-ordered aggregates which occupy the places of the elements g and g' and g -< g', then also Fg  -<  Fg', then the aggregate H, arising by combination in this manner of the elements of all the aggregates Fg, is well-ordered. 
Proof. Both H and every part H1 of H have lowest elements, and by theorem B this characterizes H as a well-ordered aggregate. For, if g1 is the lowest element of G, the lowest element of Fg1 is at the same time the lowest element of H. If, further, we have a part H1 of H, its elements belong to definite aggregates Fg, which form, when taken together, a part of the well-ordered aggregate {Fg}, which consists of the elements Fg and is similar to the aggregate G. If, say, Fg0 is the lowest element of this part, then the lowest element of the part of H1 contained in Fg0 is at the same time the lowest element of H." (p.140-141)

What can I say? Same old same old. But you should not have to take my word for it. Let us look at it more closely.

What does E mean exactly?
a) we start with a well-ordered set G of elements g; g  -<  g'; 
b) if to each g we substitute a well ordered set Fg, and to each g' Fg', then we get
c well-ordered set H of well-ordered sets Fg and Fg'; Fg  -<  Fg';

In symbols:
G= {g1,g2,.....,g,g1',g2',...,g'}
H= {{Fg},{Fg'}}, or even better
H= ({Fg},{Fg'}). [Anyway, you get the idea, H is the set formed by the union of the sets Fg and Fg'.]

The proof:
i)   Both H and every part H1 of H have lowest elements, H=well ordered aggregate by theorem B.
ii)  if g1 is the lowest element of G, the lowest element of Fg1 is at the same time the lowest element of H.
iii) If, further, we have a part H1 of H, its elements belong to definite aggregates Fg, which form, when taken together, a part of the well-ordered aggregate {Fg}, which consists of the elements Fg and is similar to the aggregate G.
iv)  If, say, Fg0 is the lowest element of this part, then the lowest element of the part of H1 contained in Fg0 is at the same time the lowest element of H.

(i) and (ii) being, or so it seems to me, unproblematic, let us concentrate on the rest.

iii) H1= Fg=G; H1 subset of H;
iv)  Fg0 lowest element of Fg; lowest element of H1; lowest element of H.

We immediately see "where the shoe pinches" [this is a literal translation of the Dutch expression "waar de schoen wringt", which means simply "where the problem lies". I find the expression very... expressive].

Cantor, as usual, assumes that what he is supposed to prove, which, I remind you, was:
"the aggregate H... is well-ordered". 

To be fair, Cantor proves that H is well-ordered by constructing it. Which would be quite acceptable if he was trying to prove the existence of well-ordered sets in general. That is certainly not the case here, nor was that the case by the other proofs.
We are expecting that Cantor proves that a set possesses the property of being well-ordered when some conditions are met. To show how to construct one would not be much help in recognizing a set as a well-ordered set in a random problem setting.
We still have no idea when to prove that a set has a lowest element, nor when an element f answers to the definition of a Cantor's cut. [It concerns condition II which says that there is an element f that divides two parts of a set.].

We are still dependent on empirical sources to establish those properties without doubt.
That would certainly not be an issue if Cantor just came out and admitted it. We would then know:
1) when to recognize a well-ordered set when we see one with the help of both conditions stated at the beginning of the article;
2) how to construct a well-ordered set.

Though We would still have no idea whether a random set is a well-ordered set or not unless it is presented to us as such.

I find this conclusion, if correct, quite fascinating. It shows the limits of mathematical proofs, and their dependence on empirical knowledge, even if this knowledge can be said to be itself mathematical.

The Liar Paradox (and other beasties)
Cantor's Logic (13.2) The Segments of Well-Ordered Aggregates (continued)

Proof of C

"C. A well-ordered aggregate F is similar to no part of any one of its segments A.
Proof. Let us suppose that F' is a part of a segment A of F and F' s~ F. We imagine an imaging of F on F' ; then, by theorem A, to a
segment A of the well-ordered aggregate F corresponds as image the segment F" of F' ; let this segment be determined by the element f' of F'. The element f' is also an element of A, and determines a segment A' of A of which F" is a part, The supposition of a part F' of a segment A of F such that F' s~ F leads us consequently to a part F" of a segment A' of A such that F" s~ A. The same manner of conclusion gives us a part F'" of a segment A" of A' such that F'" s~ A'. Proceeding thus, we get, as in the proof of theorem B, an infinite series of segments of F which continually become smaller :
A > A' > A"... A(v) . A(v+1)...,
and thus an infinite series of elements determining these segments :
f >- f' >- f" ... f(v) >- f(v+1)
in which is no lowest element, and this is impossible by theorem A of section 12. Thus there is no part F' of a segment A of F such that F' s~ F." (p.145-146)

i)    Let us suppose that F' is a part of a segment A of F and F' s~ F.
ii)   We imagine an imaging of F on F';
iii)  then, by theorem A, to a segment A of the well-ordered aggregate F corresponds as image the segment F" of F';
iv)   let this segment be determined by the element f' of F'.
v)    The element f' is also an element of A, and
vi)   determines a segment A' of A of which F" is a part.
vii)  The supposition of a part F' of a segment A of F such that F' s~ F leads us consequently to a part F" of a segment A' of A such that F" s~ A.
viii) The same manner of conclusion gives us a part F'" of a segment A" of A' such that F'" s~ A'.
ix)    Proceeding thus, we get, as in the proof of theorem B, an infinite series of segments of F which continually become smaller :
A > A' > A"... A(v) . A(v+1)...,
x)    and thus an infinite series of elements determining these segments :
f >- f' >- f" ... f(v) >- f(v+1)
in which is no lowest element, and
xi)  this is impossible by theorem A of section 12.
xii)  Thus there is no part F' of a segment A of F such that F' s~ F.

The first sentence (i) is easy to picture in our mind, and even on paper.
_______________________ F
_____________ A
________ F'

Which makes (ii) quite a conundrum:

F' s~ F?

"We imagine an imaging of F on F'". How could we do that, knowing what we know? In this case, let us forget what we know, which means that our starting situation, however strange that may sound after (i), looks likes this:

_____________________... F
_____________________... A
_____________________... F'

Now we can imagine F' s~ F. But we are far from done with the imagining part.

(iii) says we get then F" as a segment of F'. Does it look familiar? Yes, Cantor is up to his old trick of creating extra elements by almost each step. Where does this F" come from, pray?
Theorem A, which I had just analyzed previously, and which did not seem to prove anything at all, says:

"A. If two similar well-ordered aggregates F and G are imaged on one another, then to every segment A of F corresponds a similar segment B of G, and to every segment B of G corresponds a similar segment A of F, and the elements f and g of F and G by which the corresponding segments A and B are determined also correspond to one another in the imaging."

That is supposed to explain that F" is similar to A. Does it indeed? There is no law that prescribes such a relation. F" could be a segment of F' without being an image of A on F. Assuming this is in fact already assuming the solution, which is of course the contradiction of such a statement.
Theorem A, like any good theorem, speaks of an arbitrary segment A, while we are here dealing with specific segments such that F' is a part of A. Assuming that A s~ F", while the latter is itself a part of F', is already assuming that a segment can be similar to its part. This is an illegal move.
Remember, we want to prove that "A well-ordered aggregate F is similar to no part of any one of its segments A", and we want to do that by assuming that it is false.
Isn't that the whole point, showing that the assumption F' s~ F, and therefore F' s~ A, leads to a contradiction? Yes, but not by assuming something of which we already know, by theorem A, that it would indeed lead to a contradiction. If that is not clear then think of the following.
You want to prove that F is not similar to F'.
You then take theorem A as your witness, and declares that F' is not similar to A', which is not similar to F. What have you then proven except that theorem A should be considered as valid? Where does that leave theorem C, the one which we want to prove? If showing that theorem A is applicable to the situation at hand is in itself sufficient to prove theorem C, why do we need C at all?

Let us plodder on.

Starting from (iv) Cantor's cut is is put to bear on the whole argumentation, just like it did with theorem B. Are both cases similar then?
Cantor would certainly like us to think so, even though he gives us absolutely no reason to. Why (vii) is valid remains an unspoken mystery. In the discussion of theorem B the creation of an equality which kept returning at each lower level justified somehow the impossibility of an infinitely small part of being similar to that which it is a part of. Here, the infinite series is simply inexplicable. By referring to the analogy of the previous proof Cantor suggests that the same procedure can be legally applied, but that remains pure suggestion, a rhetorical trick as it were.

The Liar Paradox (and other beasties)
Cantor's Logic (13.1) The Segments of Well-Ordered Aggregates

Proof of A
"A. If two similar well-ordered aggregates F and G are imaged on one another, then to every segment A of F corresponds a similar segment B of G, and to every segment B of G corresponds a similar segment A of F, and the elements f and g of F and G by which the corresponding segments A and B are determined also correspond to one another in the imaging.
Proof. If we have two similar simply ordered aggregates M and N imaged on one another, m and n are two corresponding elements, and M' is the aggregate of all elements of M which precede m and N' is the aggregate of all elements of N which precede n, then in the imaging M' and N' correspond to one another. For, to every element m' of M that precedes m must correspond, by [section] 7 an element n' of N that precedes n, and inversely. If we apply this general theorem to the well-ordered aggregates F and G we get what is to be proved." (p.143-144)

A sound quite straightforward, so why does it need a proof?
The proof starts with an IF THEN statement. Let us look at it more closely:
i)   If we have two similar simply ordered aggregates M and N imaged on one another, 
ii)  m and n are two corresponding elements, and 
iii) M' is the aggregate of all elements of M which precede m and 
iv)  N' is the aggregate of all elements of N which precede n,

What is the IF clause exactly? It looks like a complex clause containing diverse statements linked by AND. But when we look at (ii-iv) we see that there are more like added precisions to (i) than extra conditions. Which means that we can ignore them for the time being and consider (i) as if it were a simple clause, to which we can append

v)  then in the imaging M' and N' correspond to one another.
vi) to every element m' of M that precedes m must correspond, by [section] 7 an element n' of N that precedes n, and inversely

Sentence (vi) also sounds like a specification of the then clause, as indicated by the adverb "for" linking (vi) to (v) by a causal relation. That leaves the conclusion which, very surprisingly, appeals, just like the previous statement, to other arguments to be found in section 7.

vii) if we apply this general theorem to the well-ordered aggregates F and G we get what is to be proved.

It looks like this proof does little more than specifying what we should understand under the imaging of one well-ordered set by another, and then refers the reader to section 7. But where in section 7 are we supposed to look?
Taking (vi) as my guide I realize that what I wrote about section 7 is of no help at all. I must have skipped an essential part in my analysis. Something to do with elements preceding each other in the original set as well as in its image. But then I must admit that I cannot find anything in section 7 that goes beyond the description of what the imaging of one well-ordered set by another means. Clearly, there is nothing to be gained from section 7 that has not been stated as clearly in (A) and in its so-called proof. 
The reference to section 7 could therefore be left purely and simply out, without affecting the validity of the proof neither positively nor negatively. In other words, what you read here is all there is to get.
"the general theorem" Cantor refers to simply does not seem to exist, unless of course I am gravely mistaken. All we have are descriptions of well-ordered sets and their relations to each other. Most of those relations are still veiled in a heavy mist, which I hope will clear up soon.

[s~ means "similar". Equivalence symbol with an underscore.
>-  means "follows" and is the inverse of -<]

Proof of B
"B. A well-ordered aggregate F is not similar to any of its segments A.
Proof. Let us suppose that F s~ A, then we will imagine an imaging of F on A set up. By theorem A the segment A' of A corresponds to the segment A of F, so that A' s~ A. Thus also we would have A' s~ F and A' < A. From A' would result, in the same manner, a smaller segment A" of F, such that A" s~ F and A" < A' ; and so on. Thus we would obtain an infinite series
    A > A' > A"... A(v) > A(v+1)...
of segments of F, which continually become smaller and all similar to the aggregate F. We will denote by f, f', f", . . . , f(v), ... the elements of F which determine these segments ; then we would have
f >- f >- f" >-... >- f(v) >- f(v+1)...
We would therefore have an infinite part 
of F in which no element takes the lowest rank. But by theorem A of section 12 such parts of F are not possible. Thus the supposition of an imaging F on one of its segments leads to a contradiction, and consequently the aggregate F is not  similar to any of its segments.
Though by theorem B a well-ordered aggregate F is not similar to any of its segments, yet, if F is infinite, there are always other parts of F to which F is similar. Thus, for example, the aggregate
is similar to every one of its remainders
Consequently, it is important that we can put by the side of theorem B the following  [C and its proof]:" (p.144-145)

This is quite a long proof, so it is even more important to divide it in digestible chunks. Let us take the first part, leaving the issue of infinite sets for later.

i)   Let us suppose that F s~ A, 
ii)  then we will imagine an imaging of F on A. 
iii) By theorem A the segment A' of A corresponds to the segment A of F, so that A' s~ A. 
iv)  Thus also we would have A' s~ F and also A' < A. 
v)   From A' would result, in the same manner, a smaller segment A" of F, such that A" s~ F and A" < A' ; and so on. 
vi)  Thus we would obtain an infinite series     A > A' > A"... A(v) > A(v+1)... of segments of F, which continually become smaller and all similar to the aggregate F. 
vii)  We will denote by f, f', f", . . . , f(v), ... the elements of F which determine these segments ; 
viii) then we would have f >- f >- f" >-... >- f(v) >- f(v+1)...
ix)   We would therefore have an infinite part     (f,f',f",...,f(v),...) of F in which no element takes the lowest rank. 
x)    But by theorem A of section 12 such parts of F are not possible. 
xi)   Thus the supposition of an imaging F on one of its segments leads to a contradiction, and 
xii)  consequently the aggregate F is not similar to any of its segments.

First, we must keep clearly in mind what the objective is: to prove that "A well-ordered aggregate F is not similar to any of its segments A."
Cantor chooses as strategy a reductio, he assumes in (i) the opposite of what he wants to prove, and hope to unveil a contradiction. Will it be a reductio à la Euclid or à la Cantor?
We are confronted at (iii) by the usual Cantorian gambit: he introduces, for no apparent reason, an extra element in the argumentation, the segment A'. Instead of investigating the relation between a set and its one segment, we are now analyzing one set F and two segments A and A'.
(iii) tells us that as a segment A' is similar to A.
(iv) is the real start of the reductio. It states that F is similar to one of its part, A'. But then we get a very unexpected assertion according to which A' < A. Where does this come from? I checked it with the German text, and it says the same. so l will try and make sense out of this incongruity. Maybe the rest of the proof will shed some light on this strange issue.
(v) and (vi) sound like the signs of a mental breakdown. Could it be one of cantor's famous depressions? Instead of an explanation we are treated to an infinite regression, where the same mysterious inequality shows up again and again: A" < A', and, to make matters worse, and repeat Cantor's incantation, "and so on". We must not forget that we are still under the influence of the reduction stating that F is equal to its parts. So there we have ever slinking segments that threaten to repeat the "Honey I shrunk the kids" scenario, while at the same time having to consider those disappearing segments as being still similar to F which, lucky for us, has not changed sides nor size.
(vii) is crucial in that the infinite series of ever diminishing segments, all similar to F, also point at the existence of an equally infinite number of elements f which determine those segments. Remember Cantor's cut? The f element is that point which in fact divides, but also links different parts together. Every two parts or segments have of course to meet or split somewhere. When we are dealing with normal segments f can be considered as the highest element of the preceding segment, or the lowest one of the following segment. But since all those segments must be, within the reductio argument, considered as similar to f, all their corresponding points will therefore also be similar and there will be no point f which could be considered as the lowest of this series in F. Since Theorem A of section 12 states that "Every part F1 of a well-ordered aggregate F has a lowest element", we must conclude that such an infinite series is impossible in F, and that therefore the assumption (i) that F s~ A must be false.

We still do not know why A' is supposed to be less than A. After all, it is what starts this whole shrinking mess, so it would be nice to know if we could have done something about it. Before looking any further I would like first to assume that it is correct, and review the proof as such.

The question that immediately comes to (my) mind is: why does Cantor need an extra segment A' for his argumentation, not to speak of a whole infinite series? 
We have already seen a failed example of such a strategy in Cantor's Logic (12), [Proof of A], will we see here the same inability to make it work?
We realize that this time Cantor's strategy seems to have evolved. The idea of an ever shrinking series of segments is a powerful one. It brings the point home, which could certainly not be said of the previous example mentioned. We also understand why it is essential for A' to be less than A, and that the same inequality be repeated at each step.
Does that mean that the introduction of an extra element, segment A' is justified? What did cantor really prove with his shrinking segments?
Without the inequality, an extra segment A' would have made the proof, just like before, unnecessarily complicated. Nothing could be proven of A' that could not have been proven of A. Which means that nothing could have been proven at all.
You can no more prove that F is smaller than A, than you can prove that A is similar to A. Either you see it, or you don't.
I am sure that this claim, especially concerning the first part, will not sound obvious to all. Allow me to elaborate.
You can establish, with one to one correspondence for instance, that one set is smaller than, larger than, or identical to a second set.
If you keep elements over from one set, you will say that it is larger than the other one. You will call them identical if you keep no element at all. Isn't that a proof?
Certainly. An empirical proof. What Cantor is trying to do is to dispense with empirical proofs and replace them with abstract ones. And that is the reason why he fails each time.

An extra segment, or even a whole infinite series of them, would therefore make no difference. They had to be shrinking away to become acceptable. The difference in size becomes obvious, just like in a direct empirical observation. Which in a way it was, since A' had been declared as less than A from the very beginning.

Mind you, the validity of this infinite series has not been established, just its convincing power when considered as legitimate.
See, the question of why A' is assumed to be less than A has received a strategic answer: it was necessary to make the proof stand. This still does not justify such an opportunistic choice. I am afraid that I could not find a single justification for this assumption and can therefore only admire Cantor's rhetorical ingenuity in sneaking it in to achieve his goal.
Cantor proved his point the only way he could: by cheating.

Rests his claim that infinite sets can be similar to the remainder of a segment. Not really a surprising claim, and something which he still has to prove.

The Liar Paradox (and other beasties)
Cantor's Logic (13.3) The Segments of Well-Ordered Aggregates (end)

Proof of D

"D. Two different segments A and A' of a well-ordered aggregate F are not similar to one another.
Proof, If A' (p.146)

I can be short about this proof. Theorem B was based on an unproven inequality that repeated itself at lower levels ad infinitum. It cannot therefore be used as a justification for this theorem. Also, D sounds like a tautology to me:
If A and A' are not only distinct but also different from each other, how could they ever be similar? And how could you ever prove that except by looking at the two segments and seeing that they are indeed not similar?


Proof of E

"E. Two similar well-ordered aggregates F and G can be imaged on one another only in a single manner.
Proof. Let us suppose that there are two different imagings of F on G, and let f be an element of F to which in the two imagings different images g and g' in G correspond. Let A be the segment of F that is determined by f, and B and B' the segments of G that are determined by g and g'. By theorem A, both A s~ B and A s~ B', and consequently B s~ B', contrary to theorem D." (p.146)

It is starting to get finally interesting. This theorem claims to prove a mysterious property that has been alluded to already at the beginning of the article, without any concrete clues as to its meaning. We should look forward to the proof that some sets can be imaged on each other in infinite ways, but we will first start with the less problematic cases. I would like also to point out at my remarks in Cantor's Logic (4), where I posed the question of the existence of a to the bth power. It also concerned the question whether a set, once it had been interpreted in a certain way, did not exclude all other possibilities.

________f_______ F

________g__g'_____ G
 B       B'

Constructing the case according to the instructions yields an obvious solution. If, starting with F and G similar, we add not only g as similar to f, but also g' as different from f, then G changes its form and becomes different from F. Cantor wants to prove such an obvious empirical fact, which means denying its truth at first. A simple description of what happens when we assume such a contradiction, f similar to g as well as to g', is not enough. We must act like we are dealing only with an abstract case where empirical intuition plays no role.
The problem we are facing now is that theorem E is also incomprehensible when we take away our empirical intuition. You cannot prove similarity, and therefore, you cannot prove dissimilarity either. You have to perceive it or take it as a given.
Furthermore, just like D, E looks suspiciously like a tautology gone wrong. Cantor first assumes that F and G are similar, then he investigates the case where F is imaged on G, with the expected counterpart of g corresponding to f. Then he assumes an imaging of F on G, yielding g', but without the corresponding f' in f. Which is in flagrant contradiction with the initial assumption of similarity.

Cantor has therefore not proven E, but a variation on E which could be formulated as follows:

E'. Two similar well-ordered aggregates F and G cannot be imaged on one another in different manners if the imaging is not reciprocal.

Cantor, who loves creating new elements to build his proofs, should have considered creating an extra element, f' in F, to correspond to g'. However, whether that would have proven that two similar sets can be imaged on each other in different ways is a question that remains open.


Proof of F

"F. If F and G are two well-ordered aggregates, a segment A of F can have at most one segment B in G which is similar to it.
Proof. If the segment A of F could have two segments B and B' in G which were similar to it, B and B' would be similar to one another, which is impossible by theorem D." (p.146)

_________A_____________________A_____________ F

_________B________B'__________________A'_____________ G

F is merely a reformulation of E, substituting images for segments.


Proof of G

G. If A and B are similar segments of two well-ordered aggregates F and G, for every smaller segment A'
The proof follows from theorem A applied to the similar aggregates A and B." (p.146-147)

Should we take Cantor on his word? That would not be very advantageous, since we still have to find a single theorem that can stand up to our criticism.
At the risk of repeating myself ad nauseam, I will point at the impossibility of proving formally the similarity of two elements. Even though I must also acknowledge the fact that what we are dealing with here is not so much similarity as its presence. We may no be able to prove that A and B are similar unless it has been somehow given, but we might, once it has been given in one situation, infer from that fact that it has to be present in another. This is a subtle, but nevertheless fundamental distinction.
Theorem A, even though its status turned out to be uncertain, was not shown as being invalid, but simply as being much too obvious. I have the strong impression that G can claim to be as obvious by itself, without needing the help of theorem A.


I will skip H and I because they look too much like the case of G.


Proof of K

"K. If for any segment A of a well-ordered aggregate F there is a similar segment B of another well-ordered aggregate G, and also inversely, for every segment B of G a similar segment A of F, then F s~ G.
Proof. We can image F and G on one another according to the following law : Let the lowest element f1 of F correspond to the lowest element g1 of G. If f  >-  f1 is any other element of F, it determines a segment A of F. To this segment belongs by supposition a definite similar segment B of G, and let the element g of G which determines the segment B be the image of F. And if g is any element of G that follows g1 it determines a segment B of G, to which by supposition a similar segment A of F belongs. Let the element f which
determines this segment A be the image of g. It easily follows that the bi-univocal correspondence of F and G defined in this manner is an imaging in the sense of section 7. For if f and f' are any two elements of F, g and g' the corresponding elements of G, A and A' the segments determined by f and f', B and B' those determined by g and g', and if, say,
f' -< f,
By theorem H, then, we have
B' <  B,
and consequently
g' -<  g." (p.147-148)

K sounds like a genuine theorem to me. It infers the similarity of two sets from the similarity of their segments, which is part of the given. We are here reminded of the distinction made previously (Proof of G), of the proof of similarity vs the proof of the presence of similarity.
The appeal to theorem H poses a small problem. I did not think H (and I), were worth the extra effort. But now it seems like Cantor considers them as steeping stones in his analysis. Luckily for me, Cantor affirms, talking about theorem H: "The proof follows from the theorems F and G." That allows me to claim that K, as far as I am concerned, does not need the help of those theorems at all. They all rely on similarity as a given, or on the more logical inference of similarity on the basis of similarity elsewhere. 


Proof of L

"L. If for every segment A of a well-ordered aggregate F there is a similar segment B of another well-ordered aggregate G, but if, on the other hand, there is at least one segment of G for which there is no similar segment of F, then there exists a definite segment B1 of G such that B1 s~ F.
Proof. Consider the totality of segments of G for which there are no similar segments in F. Amongst them there must be a least segment which we will call B1. This follows from the fact that, by theorem A of section 12, the aggregate of all the elements determining these segments has a lowest element ; the segment B1 of G determined by that element is the least of that totality. By theorem I, every segment of G which is greater than B1 is such that no segment similar to it is present in F. Thus the segments B of G which correspond to similar segments of F must all be less than B1, and to every segment B  <  B1 belongs a similar segment A of F, because B1 is the least segment of G among those to which no similar segments in F correspond. Thus, for every segment A of F there is a similar segment B of B1, and for every segment B of B1 there is a similar segment A of F. By theorem K, we thus have
F s~ B1." (p.148-149)

How are we to understand L? 



Let us call D the segment of G with no counterpart in F. We immediately realize that it is in fact B1, which corresponds to F and which therefore make B1 s~ F. But since we are not allowed to rely on geometrical constructions, we have to find a formal proof for our result. 

i)    Consider the totality of segments of G for which there are no similar segments in F.
ii)   Amongst them there must be a least segment which we will call B1
iii)  ... the aggregate of all the elements determining these segments has a lowest element ; 
iv)   The segment B1 of G determined by that element is the least of that totality. 
v)    By theorem I, every segment of G which is greater than B1 is such that no segment similar to it is present in F. 
vi)   Thus the segments B of G which correspond to similar segments of F must all be less than B1, and 
vii)  to every segment B  <  B1 belongs a similar segment A of F, 
viii) because B is the least segment of G among those to which no similar segments in F correspond. 
ix)   Thus, for every segment A of F there is a similar segment B of B1, and 
x)    for every segment B of B1 there is a similar segment A of F. 
xi)   By theorem K, we thus have F s~ B1.

The first 2 statements sounded first very strange to my ears. The least of those segments will necessarily have the length of a unity, 1. But then I realized that that was exactly what Cantor was looking for: the point which marks the passage from F to G, or what he called until now the f element. I call it Cantor's cut. He calls it here B1.

By (v) B1 becomes therefore something like Cantor's cut, everything below may be considered as being also a part of F while everything above will be seen as belonging only to G. Which makes the conclusion (xi) very unsurprising.

What can we learn from this proof? The most striking feature, as far as I am concerned, is the fact that all seems to be based on the existence of the point B1. How is its existence established?
The term "segment" which is used throughout the article starting from section 13, sounds certainly like a geometrical concept:

"If f is any element of the well-ordered aggregate F which is different from the initial element f1, then we will call the aggregate A of all elements of F Which precede f a "segment (Abschnitt) of F," or, more fully, "the segment of F which is defined by the element f" (p.141)
B1 depends therefore on our spatial intuition and experience [I add this last point to avoid any Kantian prejudices]. We have as given:
- all elements of F have their image on G, but not vice versa. Some elements of G have no counterpart in F.
- Both sets are well-ordered and have therefore a lowest and highest element.
- The point of passage from one set to the other will necessarily be the highest of F and the lowest of G. Unless of course, we decide to append G at the beginning of G, in which case the relation between the elements of both sets would be inversed.
- However we look at it, the difference between F and G amounts to the segment starting from the point B1 and then either going up or down according to where G lies relative to F.
- The segment B1 which, according to the definition, is the segment determined by the point B1, contains therefore all the points having their counterpart in F, and can safely be considered as similar to F.

The beauty of this reasoning is that we do not seem to need any drawing to arrive at this conclusion. Still, the whole proof is geometrical in nature and based on our spatial intuition and experience.


Proof of M

"M. If the well-ordered aggregate G has at least one segment for which there is no similar segment in the well-ordered aggregate F, then every segment A of F must have a segment B similar to it in G." (p.149)

My first reaction to M was quite surprising to me: M is certainly not obvious, which was the case of all the theorems until now. That seemed like a welcome change. 
Because I tried to prove the theorem without first reading Cantor's solution, I quickly realized that the change was probably an illusion.
First, we must of course exclude the obvious case of the unity segment, which played an essential part in the previous proof. Then we have to discount all the sets that are greater than F since they could not be, by definition similar to F. We then have all sets whose value lies between 1 and F, F being smaller than G.
These two points make it obvious that G has to be greater than F. Which should have been obvious form the first part of the statement:
"If the well-ordered aggregate G has at least one segment for which there is no similar segment in the well-ordered aggregate F".
The segment talk is a little confusing for a non-mathematician like me. I would have preferred something like:
"If G is greater than F", to make things nicely simple. But apparently, that is asking too much of mathematicians and logicians. Where would be the mystery if they all talked like this?
Once it has been established that G is greater than F, then the second part, as Gödel would say, "leaps to the eye". Of course "every segment A of F must have a segment B similar to it in G". After all, F will be similar to that part of G which is not greater than itself.
This mental argumentation I just presented relies on spatial/geometrical experience and linguistic analysis. In principle, anybody with a minimum age and schooling should be able to arrive at the same conclusion.
What does Cantor have to say about this proof?

"Proof. Let B1 be the least of all those segments of G for which there are no similar segments in F. If there were segments in F for which there were no corresponding segments in G, amongst these, one, which we will call A1, would be the least. For every segment of A1 would then exist a similar segment of B1, and also for every segment of B1 a similar segment of A1. Thus, by theorem K, we would have
B1 s~ A1
But this contradicts the datum that for B1 there is no similar segment of F. Consequently, there cannot be in F a segment to which a similar segment in G does not correspond." (p.149)

i)    Let B1 be the least of all those segments of G for which there are no similar segments in F. 
ii)   If there were segments in F for which there were no corresponding segments in G, amongst these, one, which we will call A1, would be the least. 
iii)  For every segment of A1 would then exist a similar segment of B1, and also 
iv)   for every segment of B1 a similar segment of A1
v)    Thus, by theorem K, we would have B1 s~ A1
vi)   But this contradicts the datum that for B1 there is no similar segment of F. 
vii)  Consequently, there cannot be in F a segment to which a similar segment in G does not correspond.

The first statement shows immediately how different Cantor's approach is from mine. Instead of excluding the unit segment, (i) turns it into an essential part of the proof.
(ii) starts, once again a reductio process. It assumes the opposite of M, namely that F is larger than G, because it has parts which have no counterparts in G. A1 is the logical counterpart of B1, the least element of all segments concerned.

The situation becomes very complicated after that, and we will have to tread very carefully to unravel the puzzling... puzzles Cantor has created out of a very simple problem.

The bewilderment hits immediately with (ii): "If there were segments in F for which there were no corresponding segments in G, amongst these, one, which we will call A1, would be the least." What can it possibly mean?
This is, within the reductio, the counterpart of the part which is not similar to G, and which starts with B1. We are assuming therefore that F greater than G.
(iii) and (iv) are more problematic. If we are dealing with a reductio, that F is greater than G, then (ii) makes sense. But certainly not (iii) and (iv). Not both statements can be used at the same time, F > G, and G > F. And that is the only reason why there could be segments A1 and segments B1 at the same time which are similar.
Because an impossible situation has been created, the reductio looks like it is successful. What makes this plausible is the fact that G being also greater than F, in contradiction with the reductio assumption, leads to a conclusion contrary to this last assumption.

I will not ask you if your head hurts, because just thinking about "heads" makes mine spin even more wildly!
I definitely prefer my approach!


Proof of N

"N. If F and G are any two well-ordered aggregates, then either :
(a) F and G are similar to one another, or
(b) there is a definite segment B1 of G to which F is similar, or
(c) there is a definite segment A1 of F to which G is similar ;
and each of these three cases excludes the two others." (p.150)

Obviously, (b) and (c) are just a fancy way of saying " G greater than F" or "G less than F". I am really curious as to how Cantor thinks he can solve such a "theorem" which sounds, if anything ever did, like one of the primitive intuitions on which our capacity of reasoning itself is based. 

"Proof. The relation of F to G can be any one of the three :
(a) To every segment A of F there belongs a similar segment B of G, and inversely, to every segment B of G belongs a similar one A of F;
(b) To every segment A of F belongs a similar segment B of G, but there is at least one segment of G to which no similar segment in F corresponds ;
(c) To every segment B of G belongs a similar segment A of F, but there is at least one segment of F to which no similar segment in G corresponds.
The case that there is both a segment of F to which no similar segment in G corresponds and a segment of G to which no similar segment in F corresponds is not possible ; it is excluded by theorem M.
By theorem K, in the first case we have
F s~ G.
In the second case there is, by theorem L, a definite segment B1 of B such that
B1 s~ F;
and in the third case there is a definite segment A1 of F such that 
A1 s~ G.
We cannot have F s~ G and F s~ B1 simultaneously, for then we would have G s~ B1, contrary to theorem B; and, for the same reason, we cannot have both F s~ G and G s~ A1 . Also it is impossible that both F s~ B1 and G s~ A1, for, by theorem A, from F s~ B1 would follow the existence of a segment B'1 of B1 such that A1 s~ B'1. Thus we would have G s~ B'1, contrary to theorem B." (p.150-151)

Cantor's habit of creating an extra level as something of a smoke screen [I am not attributing to him nefarious intentions, just pointing at how it works in an argumentation] is here also quite evident.
We can wonder how the points a, b and c in the proof are any different from those in the theorem. Apparently, in Cantor's mind, what can be said of segments cannot be said directly of their sets. Once we reject this assumption then we realize that Cantor's so-called proof is nothing else but a repetition of what had already been said in the theorem.
And, as I said already, that should not surprise us. I would have been more surprised if Cantor had somehow proven this theorem!


Proof of O [sorry, not the story]

O. If a part F' of a well-ordered aggregate F is not similar to any segment of F, it is similar to F itself.
Proof. By theorem C of section 12, F' is a well-ordered aggregate. If F' were similar neither to a segment of F nor to F itself, there would be, by theorem N, a segment F'1 of F' which is similar to F. But F'1 is a part of that segment A of F which is determined by the same element as the segment F'1 of F'. Thus the aggregate F would have to be similar to a part of one of its segments, and this
contradicts the theorem C." (p.151)

This is again a cryptic statement that relies solely on the ambiguity of the term "part" which has the double meaning of real part or proper subset, and also of the whole set itself. 
Solving the linguistic ambiguity renders the proof rather superfluous.

The Liar Paradox (and other beasties)
Cantor's Logic (14.0) The Ordinal Numbers of Well-Ordered Aggregates

"Members and non-members only" (Desperado, 1996)

Two ordinal sets A and B would look like that. At least, that is how I see them.

A={1st, 2nd, 3rd, 4th, 5th}
B={1st, 2nd, 3rd}
a and b being the ordinal numbers of A and B, we would have:
a=5, b=3.          
Accordingly a+b = (A,B)o, or the ordinal number of the union set (A,B).

What does that mean? Are we even allowed to add a and b? How could we do that?
If we consider ordinal numbers just like natural numbers then A+B would become C={1,2,3,4,5,6,7,8}. But that would mean that we would have first to abandon the ordinal property of the elements of B, turning them into cardinal or natural numbers before adding them to A. And once we have finished the adding operation, we would consider the added numbers as ordinal numbers again. A very strange procedure to say the least.
Ordinality seems therefore to be a contingent property that can, and most importantly, must be put aside anytime we wish to perform any kind of calculations. Such a view would be very strange and unreal. 
Furthermore, rules of precedence always denote a certain subjective perspective. There are no objective rules of precedence because of the simple fact that an infinite number of events happen at any given time, and only the choice of the observer can bring order to this natural chaos.
Turning ordinality on and off, would therefore be impossible without new precedence rules which are independent of purely objective calculations.
Imagine combining two school classes together, each with its own grading list. Even if we assume the same grading method for both classes, there is no way to predict in the abstract what the new final list will be. A student first in one class can turn out to be the fourth in the combined class. And even if we can be sure that the second of the same class will be higher (lower is better) in the new list than the first already mentioned, he might turn out to be the sixth, or even the last one of the class, with all his unlucky friends trailing at the end of the new list.

Arithmetic operations with ordinal numbers seem therefore meaningless.

What is Cantor then trying to prove in The contributions?

The Liar Paradox (and other beasties)
Cantor's Logic (14.1) The Ordinal Numbers of Well-Ordered Aggregates (end)

This section seems only meaningful if one believes in the legitimacy of ordinal numbers, something Cantor, as far as I am concerned, was certainly not able to assert beyond doubts. When I look at the different arithmetic operations to which ordinal numbers can be submitted, I only see natural numbers, and am therefore inclined to consider the whole section as superfluous. Who needs rules about addition or subtraction of natural numbers?
Still, maybe the way multiplication is presented, can be an illustration of my misgivings.

Definition of multiplication (p.154):
"If we substitute for every element g of the aggregate G of type b an aggregate Fg of type a, we get, by theorem E of section 12, a well-ordered aggregate H whose type is completely determined by the types a and b and will be called the product a. b:
(5) Fgo=a, [ordinal Fg=a]
(6) a.b= Ho." [ordinal H]

It still remains a mystery how the substitution would take place.
The definition says that we are substituting a set to an element. How does it work?
The type of a set is its ordinal number, which is in fact nothing else but the number of its elements considered in their order of precedence. Let us say type b=3 while type a=5. We would then have 3 elements g of G, which would be all replaced, one by one. by a set of 5 elements. We would then have instead of

Voilà! Mystery solved! We have explained the multiplication of ordinals. Didn't we?
Well, all I see here is the multiplication of one number by another making use of set notation. But where is the order of precedence now?
In the new H all the elements look alike and we would be hard put to distinguish a first from a second or a third. And if we do, we would then have to specify if the order of precedence concerns a subset in its totality, and/or each individual member with each subset. Is the first in the first subset necessarily the first in the whole set, and so on? That would not be fair to the other members which in the old set had a better rating.

Already in section 8 Cantor had recognized the problem. Here is what he said about that.

"Out of two ordered aggregates M and N with the types a and b we can set up an ordered aggregate S by substituting for every element n of N an ordered aggregate Mn which has the same type a as M, so that
(3) ordinal Mn = a;
and, for the order of precedence in
(4) S = {Mn}
we make the two rules :
(1) Every two elements of S which belong to one and the same aggregate Mn are to retain in S the same order of precedence as in Mn ;
(2) Every two elements of S which belong to two different aggregates Mn1 and Mn2 have the same relation of precedence as n1 and n2 have in N." (p.120)

This is the case of multiplication just handled a while ago. What else does it tell us?

S (previously H) is the product of M (previously F) and N (previously G). That we already knew.
Let us now look at the two rules and try to make sense of them.
For that we need the previous concrete example. To avoid any ambiguity I will use both notations each time.


Rule (1) says:
"Every two elements of S which belong to one and the same aggregate Mn are to retain in S the same order of precedence as in Mn ;"
Which elements are we talking about here? In S that could mean '1,2', but also '{1,2,3,4,5},{1,2,3,4,5}'. The second part of the rule would seem to suggest that we are dealing with the first case '1,2'.
It still sounds a little bit strange because it is so obvious. After all, we have replaced each element of G aka N by all of F aka M. Since we did not change the order of precedence during the substitution it is no surprise then to find the same order in the end. Rule (1) sounds quite superfluous to me.

Rule (2) says:
"Every two elements of S which belong to two different aggregates Mn1 and Mn2 have the same relation of precedence as n1 and n2 have in N."
This sounds to my ears exactly like the first rule and deserves therefore the same comment from me.
 But more importantly, the conclusion, and evidence, of same order of precedence comes not from the rules, but from the nature of the elements involved: the natural numbers.

Imagine that M, N and S had people instead of numbers as elements, each letter representing a distinct individual, then we would have:

S={{a,b,c,d,e}, {a,b,c,d,e},{a,b,c,d,e}}.

We immediately realize that S does not make much sense. How could the same group of people be present three times in the same set, after having been present in the first? We would therefore need more letters to form


Would the rules make more sense in such a case?
Not really, and that for the simple reason that we would not be able to read the rules of precedence directly off the letters, the way we did with the numbers.

This analysis sounded very convincing to me until I found a very annoying counterexample in Warclaw Sierpinski "Cardinal and ordinal Numbers" (1965)

1-< 2 -< 5 -< 31 -< 4 -< 9 -< 6 -< 35 -< 30 [let us call this series W, after the author's first name]

This is what the author said: 
"The set of all natural numbers, besides being ordered according to increasing magnitudes or according to decreasing magnitudes, can be ordered by assuming that a -< b if and only if number a has smaller natural divisors than number b, and in the case of an equal number of divisors — if a < b." (p.189)

Obviously the rules of precedence cannot be read directly off these numbers. So, what gives?

If we try and pinpoint the difference between this (counter)example and mine using individuals (people), we will probably see that the rules of precedence, even if they are not obvious, are still there, present in the background. A talented mathematician, not me obviously, would be able to distill those rules from the number series.
This is certainly not the case with my example.
We can therefore say that ordinal rules involving numbers rely on the rules (obvious or not) of precedence already existing between numbers. While ordinal rules involving non-numbers are contingent and completely dependent on the perspective of the writer or subject.

Which would mean that so called ordinal numbers represent only one part of what could be considered as ordinality. Something mathematicians would have no problem living with, I am sure. Why would they care what non-mathematicians think about ordinality?

Well, maybe they should, because if I am right, that would mean that there are no ordinal numbers. Just different ways of looking at (natural) numbers.

The Liar Paradox (and other beasties)
Can you prove the obvious?

This is certainly not an idle question since I myself have very often pointed at the obvious character of some statements that cantor had set out to prove as a form of criticism. It would be therefore very useful if we had a clear answer to this question.

We would of course have to define "obvious", and that is more easily said than done. What is obvious to an expert will certainly not be to a layman. So, I would like to exclude any specific knowledge from the definition, except what could be considered as common to most people of a certain age, and a minimum of education.
I think it would be wrong to aim for objective criteria because there are not any that I can see. Better to be satisfied with a dynamic definition open to criticism. Let the criteria be the, always temporary, result of a public discussion.
Also, even such a vague definition of "obvious" might turn out to be difficult to put into words. Intuitions are not always clear-cut. That does not mean they are wrong. It does mean though that we must be ready to review them critically at any time.

Let us start with the last example I treated in the previous entry Cantor's Logic (14.1). [I am not concerned here with the general validity of the argument, or its legitimacy.]

I said of rule (1): "It still sounds a little bit strange because it is so obvious. After all, we have replaced each element of G aka N by all of F aka M. Since we did not change the order of precedence during the substitution it is no surprise then to find the same order in the end. Rule (1) sounds quite superfluous to me"

How could such an affirmation be justified? To answer this question we would need first to answer the main question which is the title of this entry. But not in a general and abstract way. We would need to know if rule (1) can be shown to be superfluous, or in the contrary, if it is a meaningful rule.
That is still too vague and in dire need of concretization. What do we expect exactly from rule (1)?
Apparently the rule is there to make sure that the elements {1,2} of M reappear in the same order in S. This to avoid situations like S={{3,1,4,2,5},....}, where elements change position from one set to the other. Such a rule would only be meaningful if this would be a real danger or possibility. If it could never happen, we would be entitled to consider the rule as superfluous. That was the explicit conclusion I presented in my argumentation: "After all, we have replaced each element of G aka N by all of F aka M. Since we did not change the order of precedence during the substitution it is no surprise then to find the same order in the end." Is such a conclusion justified?
I would say that it certainly seems justified because the substitution concerns two ordered aggregates, M and S, and that therefore the internal rules of precedence are implicitly protected, since changing them would turn the sets concerned into non-ordered aggregates.
The same reasoning can be applied, I think, to rule (2).

The question is, is this argumentation applicable to all obvious statements, with the critical reservations expressed above, found in the Contributions?
Let me remark that the need of proof of the obvious can be brought back to Euclid who regarded drawings as imperfect, even if construction rules were considered as admissible as proof. In other words, it would be reasonable to point at a triangle constructed according to the instructions meant for isosceles triangles, and call it an isosceles triangle. But it would be wrong to look at a random triangle, one of which we are not sure how it was constructed, and call it isosceles simply because that is how it looks like to us.

Euclid's method might be in last instance the only reasonable way to look at Cantor's proofs. Are they obvious because of the construction principles involved, or simply because of the way they look to us?
Would we we really expect Euclid to prove that a triangle which he had just constructed according to specific instructions for isosceles triangles is indeed an isosceles triangle? How could he ever do that?

That is what I think Cantor is very often doing: trying to prove the obvious, and, almost as often, failing to do so.

The Liar Paradox (and other beasties)
Cantor's Logic (9.0) The Ordinal Type n of the Aggregate R of all Rational Numbers which are Greater than 0 and Smaller than 1, in their Natural Order of Precedence

The question of the obvious treated in the previous entry motivated me to look again at the parts I had skipped. I found in section 9 a jewel of Cantorian argumentation that certainly deserves our attention.

As the title indicate, this section concerns a specific ordinal type, called eta. [I use n because it looks like the Greek letter which is actually the equivalent of 'h' as in the Latin-American 'Jose", which is softer that the original Spanish sound, itself very similar to the German guttural 'ch" as in Bach. Modern Greeks, and I suppose therefore Ancient Greeks as well, like many Eastern Europeans, cannot pronounce the English 'h', but do not skip it altogether as the French are prone to do.
I thought these few anthropological remarks about linguistic idiosyncrasies of the European tribes might help the reader better understand their psyche.]

As usual, I will follow the text as closely as necessary and assume nothing else but what it says, as I understand it, accepting only the results obtained from the analysis of the previous sections. I deem such a "naive" lecture necessary to avoid relying on mathematical knowledge that is considered as universally valid. After all, I want to know how Cantor comes to this knowledge, less than which knowledge he finally obtained.
I would also like to point out that my analysis in no way can be used to judge of the validity of this mathematical knowledge. At least, I not being a mathematician, certainly could not. The fact that certain, or even the majority of the Cantorian proofs can be considered so far as being of dubious quality does not mean that the same knowledge could not be better founded using other methods.

What do we know about the ordinal type n?

Cantor introduced it first in section 7 (p.115): speaking of inverse ordinals, where the order of precedence is simply inversed, lower becoming higher and vice versa, denoted by *, *a being the inverse of a. He gives the example of finite sets which, because the number of their elements, their cardinal number, remains unchanged whether we look at them from one end or the other. He then mentions the case of "the type of the aggregate of all rational numbers which are greater than 0 and less than 1 in their natural order of precedence." We will learn later that R has neither a first nor a last element, still, its cardinal number remains the same whether we look at the elements in an ascending or descending order. He also promises that it will be investigated "under the notation n".
A few lines later we learn that the n type is however different from finite types, or even transfinite types called "ordinal types", in that it can be "similar to itself in an infinity of ways," while omega or w types are similar to themselves in only one way.
I must admit that I find all these descriptions a little bit ambiguous. Types n seem on one hand to keep the same cardinal number under all circumstances, just like finite type which have their ordinal number equal to its inverse. On the other hand, we have to distinguish them from "ordinal types", and from those which can be similar to themselves in only one way.
Just like I said by the analysis of section 7, it is not all very clear and we should bravely move on.

Section 9 offers some additional information about type n.
Here Cantor reminds us of the distinction he had made in section 7 between the set R, with its "natural" order, and a special form of R, R0, which is ordered first by the sum of it nominator (p) and denominator (q), and secondly the overall value of p/q.
Cantor points at the specific property of R0: "to one and the same value of p + q only a finite number of rational numbers p\q belongs". Which makes it therefore different from the n type presented as a kind of set that can be "similar to itself in infinite ways". 
R0 on the the other hand, gets the w type. This, even though both R and R0 have the same cardinal number.
We have therefore:
R  is of ordinal type eta.
R0 is of ordinal type omega, set of all finite cardinal numbers.
R and R0 both have as cardinal number aleph zero.

Cantor then does something typical. He turns a factual description of R first into the definition of n, and then declares the whole as a theorem, which he then sets out to prove. I honestly have no idea why he does that, except maybe to make his articles look more imposing by the number of theorems he shows he can prove.

Theorem of ordinal type n: (p.124)
"If we have a simply ordered aggregate M such
(a) Mc = Z0 ; [cardinal M = aleph zero]
(b) M has no element which is lowest in rank, and no highest ;
(c) M is everywhere dense ;
then the ordinal type of M is n."

The proof is a few pages long, so I will start directly with its dissection. I refer the reader to the text p. 124-127. Instead of sentences, I will also use paragraphs to limit the number of statements to analyze. The analysis will follow the quote immediately.


i) "Because of the condition (a), M can be brought into the form  of a well-ordered aggregate of type w; having fixed upon such a
form, we denote it by M and put 
(5) M0 = (m1, m2, . . ., mv ,. . .)."

The first statement (i) follows from the definitions we have already met. To the sets with cardinal number aleph zero belong also those well-ordered set of type w.

ii) "We have now to show that
(6) M s~ R;
that is to say, we must prove that M can be imaged on R in such a way that the relation of precedence of any and every two elements in M is the same as that of the two corresponding elements in R."

This is Cantor's objective. Let's see if he can reach it.

iii) "Let the element r1 in R be correlated to the element m1 in M. The element r2 has a definite relation of precedence to r1 in R. Because of the condition (b), there are infinitely many elements mv of M which have the same relation of precedence in M to m1 as r2 to r1 in R ;"
This is of course crucial. If Cantor succeeds in showing that the same relation of precedence exists between r1 and r2 on on one side, and between m1 and m2 on the other side, the proof will be given that M s~ R.
The difficulty that Cantor sees in his way is somewhat nebulous: "there are infinitely many elements mv of M which have the same relation of precedence in M to m1 as r2 to r1 in R." What can we make of that? It does not look like a geometrical line composed out of points adjacent to each other. In such a situation there would be only one point to have the desired property, even if we had no way of pinpointing it. M looks rather like a multi-dimensional object spreading infinitely in an infinite number of directions. Why would Cantor create such an impossible situation, so different from the linear image conveyed by R?
I might be prejudiced, but I think that is because a linear presentation of M would have too easily shown that the problem has no formal solution at all. Let us try it.

R= {r1,r2,...rv...}
M= {m1,m2,...,mv,...}

How could you possibly prove that r1 precedes r2 just like m1 precedes m2? Where could you possibly get that knowledge except by observation, or because it was already given to you by definition or description?
Cantor had to reshuffle the deck in such a way that this obvious remark would play no role in the evaluation of his method.
Making the problem more complicated than it really is was a strategic imperative. Let us look at this strategy.

iv) "of them we choose that one which has the smallest index in M0, let it be mi2 and correlate it to r2."

[me: not you, you fool!
George (thawing): Man! I thought I was gonna have a heart attack!]

This Cantorian move should look familiar by now. Cantor creates an intermediary step and somehow assumes the problem solved or proven. All you need to do then is port the solution from one level to the other.
Not the implicit use of what will be known as the axiom of the choice is here the problem, but rather its late involvement. Why not use it immediately with R and M? Well, that wouldn't be much of a proof, would it?

v) "The element r3 has in R definite relations of precedence to r1 and r2 ; because of the conditions (b) and (c) there is an infinity of elements mv of M which have the same relation of precedence to m and m1 in M as r3 to r1 and r2 to R; of them we choose that -let it be mi2- which has the smallest index in M0, and correlate it to r3 . According to this law we imagine the process of correlation continued. If to the v elements
r1, r2, r3,..., rv
of R are correlated, as images, definite elements 
which have the same relations of precedence amongst one another in M as the corresponding elements in R, then to the element rv+1 of R is to be correlated that element miv+l of M which has the smallest index in M0 of those which have the same relations of precedence to
in M as rv+1 to r1,r2,...,rv in R.
In this manner we have correlated definite elements miv of M to all the elements rv of R, and the elements miv have in M the same order of precedence as the corresponding elements rv in R."

This group of statements follows logically from the premise stated in (iv), and can therefore safely be ignored, or at least skipped. With (iv) the punch line has been unveiled, the rest is just filling in the blanks.  

vi) "But we have still to show that the elements miv include all the elements miv of M, or, what is the same thing, that the series
1, i2, i3,...,iv,...
is only a permutation of the series
1, 2, 3, ...',v,..."

This should be interesting. M has suddenly been transformed into a linear set, just like R!

vii) "We prove this by a complete induction : we will show that, if the elements m1,m2,. . ., mv appear in the imaging, that is also the case with the following element mv+1."

A perfectly legal move. But we will see that Cantor has a special use in mind.

viii) "Let lambda be so great that, among the elements
m1,m2,m3,. . ., mil,
the elements
m1,m2,. . ., mv,
which, by supposition, appear in the imaging, are contained. It may be that also mv+1 is found among them ; then mv+l appears in the imaging."

This is the obvious case where there is nothing left to prove.

ix) "But if mv+l is. not among the elements 
m1,m2,m3,. . ., mil
then mv+1 has with respect to these elements a definite ordinal position in M ; infinitely many elements in R have the same ordinal position in R with respect to r1, r2, , ., rl amongst which let rl+s. be that with the least index in R . Then mv+l has, as we can easily make sure, the same ordinal position with respect to
m1,mi2,mi3,. . ., mil+s-1,
in M as rl+s has with respect to
in R. Since m1 m2 , . . . , mv have already appeared in the imaging, mv+1 is that element with the smallest index in M which has this ordinal position with respect to
m1,mi2,mi3,. . ., mil+s-1."

This really shows Cantor's genius. He simply swaps the respective roles and properties of M and R. This time it is R that is considered as an infinite multi-dimensional object. And what worked for M should obviously also work for R! The conclusion is then inescapable!

x) "Consequently, according to our law of correlation, 
mil+s = mv+1
Thus, in this case too, the element mv+1 appears in the imaging, and rl+s. is the element of R which is correlated to it.
We see, then, that by our manner of correlation, the whole aggregate M is imaged on the whole aggregate R; M and R are similar aggregates, which was to be proved."

A very nice trick indeed. Cantor took a statement that he could impossibly prove and built a complex argumentation that completely hid this fact, giving the impression that he had added one more stone to his imposing edifice.

[George: Nicely put. Not bombastic at all. I am so proud of you, son!
me: oh shut up!]

The Liar Paradox (and other beasties)
Cantor's Logic (14.2) 
The Logic of ordinal Numbers or Ordinal Numbers and Logic
George: why don't you call it (end 2)?
me: uh?
George: It is the continuation of Logic (14.1)...(end), right?
me: not really.
George: okay, whatever you say.]

Addition of ordinal numbers
It should not surprise us that Cantor makes a distinction between a+b and b+a. We have already encountered his strange arguments before when dealing with aleph zero, or the domain where human logic lays down its arms in meek submission to mysticism. [sorry, I got somewhat carried away].
Here Cantor mentions the, for him, certain fact that a set can be equal to one of its parts. He makes the distinction between a segment, which will always be smaller than the set it is a segment of, and the remainder of the set which can be equal to the whole set.
Within Cantorian Logic this makes perfect sense. Every segment will be considered as a finite subset, and therefore smaller than the mother set, while the remainder can be infinite, and therefore equal to its mother (p.153).

[George: who was that Greek dude that did the nasty with his mother?
me: Oedipus? What does he have to do with all this?
George: nothing, it just popped up in my mind, that's all.
me: why don't you go play in Cognitive Sciences if you're bored?
George: nah! There's nobody there. All those empty corridors, that was really depressing. Worse than The Shining. They all think your threads are rubbish or they don't know how to respond. Or (innocently), they don't know how to respond to rubbish?
me: I won't dignify this impertinence with an answer.
George chortles]

This ambiguous attitude towards (Euclidean) logic is perfectly clear in the following quote (y is gamma, a alpha, and b beta):
"In general a+b and b+a are not equal. On the other hand, we have, if y is a third ordinal number,
(4) (a+b)+y = a+(b+y).
That is to say :
E. In the addition of ordinal numbers the associative law always holds." (p.154)

And what works for addition, also works for multiplication.
So great are the Cantorian mysteries.

I would like to remind the reader that Cantor has not offered a single new argument or new shred of evidence for his outrageous statements since his (failed) attempts in section 6. If you remember, he tried to base the existence of Aleph Zero on the conceptual confusion of one to one correspondence as a checking method on one hand, and as a counting method on the other. Since then, he has been building all his system on this false argument. I am really curious whether he will be satisfied with this result, or whether he will present us with more compelling reasons to believe in his stratification of the Infinite. (see Cantor's Logic 6.0, and 6.1 for the relation between a set and its subsets)

Subtraction of Ordinal Numbers
Cantor does not, apparently, judging by the length of this sub-section, have much to say about this operation, but it seems important to him to establish its legitimacy as far as ordinal numbers are concerned. His justification is far from being straightforward, in fact, it can definitely be considered as convoluted. Instead of simply subtracting two allegedly ordinal numbers, Cantor begins first with justifying the existence of an element that can be considered as the difference between sets a and b, when a (alpha) is assumed to be less than b (beta).

(10) a+(b-a)=b.

Let us look at the rest of the quote:
"For if Go=b, G has a segment B such that B = a ;
we call the corresponding remainder S, and have
and therefore
(11) b-a = So.
The determinateness of b-a appears clearly from the fact that the segment B of G is a completely definite one (theorem D of 13), and consequently also S is uniquely given."

Let us start with the last sentence. Cantor confirms that the element represented by the subtraction operation b-a is, just like the segment B of G, a completely definite one. This insistence is rather strange, after all who would doubt the (mathematical) existence of the difference between two elements?
So, what is Cantor really trying to prove in those apparently innocuous few lines?
I would not know.
The rest of the argumentation does not bring any further clarification (y is gamma):
"We emphasize the following formulae ...
(12) (y+b)-(y+a)=b-a,
(13) y(b-a)=yb-ya."

How do these formulas justify the existence of ordinal numbers? They are simple algebraic formulas making use of everyday natural numbers.

Addition and Sums (plus everything else)
Cantor starts with a somewhat confusing statement:
"It is important to reflect that an infinity of ordinal numbers can be summed so that their sum is a definite ordinal number which depends on the sequence of the summands." (p.156).
The rest of the subsection is even more confusing since we have to read each statement very carefully to know whether we are dealing directly with ordinal numbers or their sums.
We have two segments, F, beta in the text, and G. F is an infinite series of ordinal numbers F1, F2,...,Fv. G "is also a well-ordered aggregate whose ordinal number represents the sum of the numbers Fv.
We have, then,
(16) F1+F2+. . . +Fv+ . . . =Go = F" (p.155)

An example would I think be very welcome at this point.
F={1,2,3,4,5,6}, [set of ordinal numbers]
G={1,3,6,10,15,21}. [set of sums of ordinal numbers]

Cantor then reminds us, as if it were necessary, that 

(17) y.(F1+F2+. . . +Fv+ . . .)=y.F1+y.F2+...+y.Fv+...

But then, since arithmetic laws do not seem to always be applicable, maybe it was necessary to remind us of the validity of the distributive law when ordinal numbers are concerned.

More important is the introduction of a new element (why am I not surprised?) as in

(18) av=F1+F2+...+Fv,

add to that

(19) av = (G1,G2, . . . Gv,)o. [the whole right expression with a single bar]

and it almost starts to make sense.
The a's represent sums of ordinal numbers, just like the G's. Right?
What is therefore the ordinal number of (G1,G2, . . . Gv)?
Let us go back to our example again.
av, since it is equal to an ordinal number, must itself be an ordinal number. It denotes any element of G.
if, as in our example
a1 will be 1, with the value of G1, which is 1
a2 will be 2, with the value of G2, which is 3,
a3 will be 3, with the value of G3, which is 6,
av will be v, with the value of v which is the sum of all preceding numbers. if v=6 as in our example
a6 will be 6, with the value of G6, which is 21

Av will therefore always be equal to the ordinal number of the corresponding element in G, and not to the value that element is indicating: a6=6, not 21.
[This is not exactly right, since av represents the sum of all the preceding numbers. But since I took the example of a finite set, it is good enough to show the logic behind Cantor's approach.]
This is a typical Cantorian argumentation where the introduction of an extra element complicates everything even when, as is the case here, it leads to valid results (given Cantor's assumptions).

What was Cantor trying to achieve? Very simply, that ordinal numbers, just like any numbers, can be summed up, and that these sums can themselves become members of a set. A simple operation has degenerated into a brain teaser because of the extra elements av.

The Liar Paradox (and other beasties)
Cantor's Logic (14.3) 

Fundamental Series (p.157 ff)

Let us look first at the result Cantor has obtained in the previous lines:
(F is beta in the text)

21) F1 = a1; Fv+1= av+1-av .

The first part, F1=a1, seems simple enough. The second part not so.
Remember our example?
F={1,2,3,4,5,6}, [set of ordinal numbers]
G={1,3,6,10,15,21}. [set of sums of ordinal numbers]

let v=2

The series a1,a2,...av, with each following element greater than the one preceding it, will be called "fundamental series".

Properties of a fundamental series
"(a) The number F is greater than av for every v, because the aggregate (G1, G2,. . ., Gv), whose ordinal number is av, is a segment of the aggregate G which has the ordinal number F"

This is a logical trap that I found very difficult to avoid. I kept taking concrete example for F and G, and found each time the opposite of what Cantor was claiming: F < G, just like (21) showed. Here is a "wrong" example as it was used, rightly, before:

F={1,2,3,4,5,6}, [set of ordinal numbers]
G={1,3,6,10,15,21}. [set of sums of ordinal numbers]

Imagine that we are now dealing with infinite sets, then we would have to speak of


In this perspective, any v will be smaller than the whole infinite set F, and any set Gv will be smaller than G.

Cantor's obscure style gives a false impression of depth.

The second property of a fundamental series sounds much more straightforward:

"(b) If F' is any ordinal number less than F, then, from a certain v onwards, we always have
av > F'."

But cantor would not be Cantor if he kept things simple.
This is his short proof:
"For, since F' < F, there is a segment B' of the aggregate G which is of type F'. The element of G which determines this segment must belong to one of the parts Gv ; we will call this part GV0. But then B' is also a segment of (G1,G2,. . ., Gv0), and consequently F' < av0. Thus
av > F'
for v >= v0."

_________________________________________ F
__________________________ F'

_________________________________________ G
__________________________ B'

Looking at the drawing, we realize how ridiculous Cantor's argumentation is. Gv0 is defined as the point that determines the segment B', or as what I called Cantor's cut. Since av0=Gv0, all the points beyond Cantor's cut will have a higher value. This a matter of definition. In fact, a linguistic analysis of (b) would have shown that a proof, even if possible, was completely superfluous:

"(b) If F' is any ordinal number less than F, then, from a certain v onwards, we always have
av > F'."

(b') If F' is any ordinal number less than F, from a certain v onwards, that is from any v greater than F', then av > F'."

So, how do you prove a tautology? You make it sound very complicated.

The Concept of Limit
This is, I think, what Cantor has been aiming at all along. He had to give the impression of a final destination that kept shifting away and always just out of reach. And I must admit, he certainly succeeded in his plan: instead of starting from the "obvious", a point in infinity that keeps moving away, he gets to it by successive "proofs" that seem to finally bring the unreachable infinite within our grasp. There is nothing wrong with the rhetorical objective, except that it is wrapped in a pseudo-mathematical blanket.

Here is the long awaited statement:

"Thus F is the ordinal number which follows next in order of magnitude after all the numbers av ; accordingly we will call it the "limit" (Grenze) of the numbers av for increasing v and denote it by Limv av..." (p.158)

I think it is better for me to stop right here. Limit is an essential mathematical concept and it would be pretentious of me to try and discuss it as I did the other concepts Cantor has used to get to this point. As I said before, the fact that Cantor botched the job of founding this concept on a secure footing does not mean that the concept of Limit should be thrown overboard. Even I know that it is much too useful for that. Whether there is a better foundation for it is an epistemological matter that only mathematicians can research. I will leave it to them.

The Liar Paradox (and other beasties)
Cantor's Logic (15.0)
The Voyage beyond Infinity
At the end of section 14 Cantor starts defining the concept of Limit. As I said, I do not intend to discuss the validity of this concept as such, nor its mathematical use. Still, Cantor uses the Limit concept to justify the further construction of his system. I must therefore look critically as his presentation of this concept, and the proofs he puts forward to justify the existence of other transfinite numbers like Aleph One. 
For that, I will start at the end of section 14 and move from there to section 15 and beyond.

"To every fundamental series {av} of ordinal numbers belongs an ordinal number Limv av which V follows next, in order of magnitude, after all the numbers av ; it is represented by the formula (22)." (p.158)
(22) Limv av= a1 + (a2-a1) + . . . + (av+1-av) + . . .

This is of course a direct consequence of formula (21) which I have analyzed in the previous entry.
Let me first remark that (22), whatever its practical utility in mathematics, is not a number, transfinite or otherwise. It is more of a construction procedure which gives a definite number as a result, once v has been defined. The number constructed will by definition be a finite number.
In contrast to (22), the following two statements seem to concern definite numbers:

(23) Limv (y + av) = y + Limv av, ;
(24) Limv y . av=y . Limv av .

But here also, unsurprisingly since (23) and (24) are theorems, we are dealing more with a schema than with definite numbers. Both statements tell us how to handle specific numbers having the forms stated.
There are no formulas for infinite numbers because that would be a contradictio in terminis. Any formula would necessarily create a definite number, which can therefore only be finite.

Cantor does not seem to be aware of this difficulty. In fact, his optimism is such that he thinks he can prove the existence of a stratified Infinite. The first step is of course the distinction between the finite and the infinite. A distinction which he has dealt with many times. He does it again here, trying for a last time to prove this crucial distinction.

Finite ordinal numbers and their proof
"two different ordinal numbers a and b cannot belong to the same finite cardinal number v." (p.159)

Again, given the definitions of ordinal and cardinal, an obvious statement that Cantor sets out to prove.
The above statement can be reformulated as follows:
Two sets with a different number of elements, and different order of precedence, cannot have the same number of elements.
How do you prove that? Cantor knows just what to do:

"For if, say, a < b and Go = b, then, as we know, there exists a segment B of G such that Bo = a. Thus the aggregate G and its part B would have the same finite cardinal number v. But this, by theorem C of section 6, is impossible. Thus the finite ordinal numbers coincide in their properties with the finite cardinal numbers."

i) For if, say, a < b and Go = b, then, as we know, 
ii) there exists a segment B of G such that Bo = a. 
iii) Thus the aggregate G and its part B would have the same finite cardinal number v. But this, by theorem C of section 6, is impossible. Thus the finite ordinal numbers coincide in their properties with the finite cardinal numbers."

Where does the conclusion of (iii) come from? It is in fact a reductio. All that rests Cantor is then to show that a set of finite numbers and one of its parts cannot share the same cardinal number. And since C of section 6 says that no finite set can be equal to one of its part, the problem is solved.

Cantor's logic can be summed up as follows:
1) create an extra element, or level
2) declare the problem obviously solved for that element or level,
3) declare the whole problem solved.

The creation of an intermediary step, of segment B, seems here to make the proof possible. But segment B only emphasizes the fact that a and b are different, and that therefore either a smaller than b, or vice versa. The logic which makes us decide that a part of a set and the set itself cannot have the same cardinal number, is only acceptable as proof if it also applies to two different and distinct sets. And if it does, then we do not need the extra intermediary step with segment B.
Cantor has therefore not proven that "finite ordinal numbers coincide in their properties with the finite cardinal numbers". It is either obvious, and therefore in no need of a proof, or it is still unproven.
This is more important than some will maybe realize. If this simple fact cannot be proven, then it becomes even more difficult to prove that infinite sets can be equal to one of their parts, and that different infinite ordinal numbers can belong to the same cardinal number, which is anything but obvious!

And that is exactly what Cantor wants us to believe:
"to one and the same transfinite cardinal number a belong an infinity of ordinal numbers which form a unitary and connected system." (p.159)
Cantor calls this system "number class Z(a)", or "the first number-class".

Cantor is ready to start the construction of his famous edifice: different levels of Infinity.

The Liar Paradox (and other beasties)
Cantor's Logic (15.1) The Numbers of the Second Number-Class Z(aleph zero )

There are many proofs in this section too, so we will deal with them one by one to avoid being overwhelmed by Cantor's logic.

Proof of A
"A. The second number-class has a least number w=Limv v.
Proof. By a> we understand the type of the well-ordered aggregate 
(1) FO = (f1,f2, ...,fv). 
where v runs through all finite ordinal numbers and
(2) fv -< fv+1,
Therefore ( section 7)
(3) w=F0o, [omega = ordinal F0]
and ( section 6)
(4) wo= aleph zero [remember, w is an ordinal number, so ordinal omega is a cardinal number]
Thus w is a number of the second number-class, and indeed the least. For if y is any ordinal number less than w, it must ( section 14) be the type of a segment of F0 . But F0 has only segments
A = (f1,f2,...,fv),
with finite ordinal number v. Thus y = v. Therefore there are no transfinite ordinal numbers which are less than w, and thus w is the least of them. By the definition of Limv av given in section 14, we obviously have w=Limv v." (p.160)

We learn something essential from this proof: Cantor has nothing new to offer.
A and its proof are in fact a mere recapitulation of what Cantor is convinced he has achieved.

1) v represents all finite ordinal numbers;
2) w is the ordinal number of v;
3) aleph zero is the cardinal number of v;
4) Limv v is what comes after v;
5) what comes after v is the end point of all finite ordinal numbers, and the start of transfinite numbers, therefore w as ordinal number, and the same as cardinal number, or aleph zero.

This last statement might sound a little bit strange. We would expect the transfinite numbers to start with aleph zero +1, or Z0+1.
But remember that Z0+1=Z0, just like w+1=w.
Limv v is therefore the highest number possible of all finite numbers [which as limit is never achieved], and the lowest of the next class, the Second Number-Class.

Why is the fact that A does not contain anything new so essential? Because it shows that in last instance Z0+1=Z0 is indeed the cornerstone of the Cantorian edifice, as I stated in 6.0. In fact, once the invalidity of that statement was established, all that has come afterwards was in a way superfluous. From a false theory anything can be proved, so that even if Cantor had somehow presented formally valid proofs, they would have still been considered as illegitimate. His failure to do so will weigh even more heavily in the final evaluation of his achievements: he not only based his system on a false principle, he could not even present formally valid proofs in most of the cases.

The Liar Paradox (and other beasties)
Cantor's Logic (15.2) The Numbers of the Second Number-Class Z(aleph zero ) (continued)

Proof of B
"B. If a is any number of the second number-class, the number a+1 follows it as the next greater number of the same number-class.
Proof. Let F be a well-ordered aggregate of the type a and of the cardinal number Z0 :
(5) Fo = a,
(6) a0 = Z0.
We have, where by g is understood a new element, 
(7) a+1=(F,g)o.
Since F is a segment of (F, g), we have
(8) a+1 > a.
We also have 
ordinal (a+1)=ao+1=Z0+1=Z0 (section 6).
Therefore the number a+1 belongs to the second number-class. Between 0 and a+1 there are no ordinal numbers; for every number y which is less than a+1 corresponds, as type, to a segment of (F, g), and such a segment can only be either F or a segment of F. Therefore y is either equal to or less than a." (p.161)

While A could be understood as a recapitulation of all past steps, B can be considered as the confirmation of the start of a new infinite series beyond all finite ordinal numbers.
We must therefore look beyond the triviality of a+1 as the follower of a, and see that, in Cantor's mind, it gives substance to the abstract idea of the second number-class.
The proof is, for a change, quite straightforward, and is summed up in "ordinal a+1=ao+1=Z0+1=Z0". It shows that we have left the plane of finite ordinal numbers, and definitely entered the realm of transfinite numbers, with w and Z0 as the door to this realm.
The problem of course is that this proof, like all others, is based on the validity of Aleph Zero.

Proof of C
"C. If a1,a2,. . ., av,. . . is any fundamental series of numbers of the first or second number-class, then the number Limv av (section 14) following them next in order of magnitude belongs to the second number class.
Proof. By section 14 there results from the fundamental series {av} the number Limv av if we set up another series b1,b2,...,bv,..., where
b1=a1, b2=a2-a1,...,bv+1=av+1-av,...
If, then, G1, G2, . . ., Gv ,
are well-ordered aggregates such that
ordinal Gv = bv,
then also
G=(G1,G2, ...,Gv,.. .)
is a well-ordered aggregate and
Limv av = Go.
It only remains to prove that
Since the numbers {b1,b2,,...,bv,,... belong to the first or second number-class, we have
cardinal Gv =<  Z0. [=< less or equal]
and thus
cardinal G =< Z0.Z0=Z0
But, in any case, G is a transfinite aggregate, and so the case Gc < Z0 is excluded.
We will call two fundamental series {av } and {a'v } of numbers of the first or second number-class (section 10) "coherent," in signs:
(9) {av}ll{a'v},
if for every v there are finite numbers l0 [lambda 0] and m0
such that
(10) a'l >av, l >= l0,
(11) am v, m >= m0." [m is the Greek mu in the text]

Normally I would skip this proof because it looks so much like the previous one. But the expression "It only remains to prove that Gc=Z0" drew my attention. It sounds so promising, doesn't it? Unfortunately, Cantor doe not have much to show for it.
Here is what his proof consists of: 

1) Gc is either equal or less than Z0.
2) we have established that that means that it is also equal to Z0.Z0.
3) Gc is therefore a transfinite number and can therefore never be less than Z0.

Nothing unexpected therefore.

Proof of D concerns the coherence relation between two limit series, and is, I think, a pure mathematical matter of which I could not say anything meaningful. I will therefore skip it.
The following proof is much more interesting.

Proof of E
"E. If a is any number of the second number-class and v0 any finite ordinal number, we have v0+a=a, and consequently also a-v0=a.
Proof. We will first of all convince ourselves of the correctness of the theorem when a=w. We have
w= (f1,f2,...,fv...),
v0= (g1,g2,...,gv0),
and consequently
v0+w= (g1,g2,...,gv0,f1,f2,...,fv...)=w
But if a > w, we have
v0+a=(v0+w)+(a-w)=w+(a-w)=a." (p.163-164)

Theorem E is in itself not really surprising. It confirms an essential property of the Cantorian transfinites: any arithmetical operation on a transfinite number (addition, subtraction, multiplication), will leave that number unchanged. The proof does not deviate from the beaten paths, still, one point deserves our attention. The last three lines of the proof are:

i) But if a > w, we have
ii) a=w+(a-w),
iii) v0+a=(v0+w)+(a-w)=w+(a-w)=a.

The question is, what kind of number is a-w? Is it a finite or transfinite number? And how could a transfinite number be greater or less than another transfinite number when anything we add to it or subtract from it gives us the same number anyway? [more on that later]
If a can be greater than w, than (iii) should not be possible. If it is possible, that means that a cannot be greater than w, and then we may rightly wonder what the whole proof is about.

As you can see, Cantorian logic has to bite its own derrière.
You cannot on one hand claim that transfinite numbers remain unchanged, and on the other hand, suspend that rule whenever it suits you.

We may not therefore, consider E as proven.

The Liar Paradox (and other beasties)
Cantor's Logic (15.3) The Numbers of the Second Number-Class Z(aleph zero ) (continued)

Proof of F
"F. If v0 is any finite ordinal number, we have v0.w=w.
Proof. In order to obtain an aggregate of the type v0.w we have to substitute for the single elements fv of the aggregate (f1,f2,...,fv....) aggregates (gv,1, v gv,2,. . . , gv,v0) of the type V0. We thus obtain the aggregate
which is obviously similar to the aggregate {fv }.
The same result is obtained more shortly as follows.
By (24) of section 14 we have, since w=Limv v,
v0w=Limv v0v.
On the other hand,
and consequently
Limv v0v= Limv v=w;
so that
v0w=w." (p.164)

Cantor is still establishing the validity, read legitimacy, of all arithmetic operations on ordinal numbers of the second number-class. Namely, the fact that the combination of finite and transfinite numbers leave the latter unchanged. The use of Limits as an alternative proof method is supposed to reinforce this legitimacy.
The problem is that the coherence of two limit series does not bring any extra layer of truth. Once you have accepted Cantor's assumption of aleph zero and his different number-classes it makes perfect sense that the limit of an infinite series will also remain the same, whatever you do with the original ordinal number; as long as it is adding, subtracting, or multiplying a finite and a transfinite number together.
Things change when both parts of the operation concern numbers of the second number-class, like it is the case in the following proof with a and w.

A new Mystery
Proof of G
"G. We have always
where a is a number of the second number-class and v0 a number of the first number-class.
Proof. We have
Limv v = w.
By (24) of section 14 we have, consequently,
(a+v0)w = Limv(a+ V0)v.
But [the preceding subscripts are a tally of the recurring expression]
      =a+ 1(v0+a)+2(v0+a)+...,+v(v0+a)
Now we have, as is easy to see,
{av+v0} || {av},
and consequently
Limv (a+v0)v=Limv (av+v0)=Limv av=aw." (p.165)

I would beg the reader to examine my analysis of this point even more critically than can be expected because I hardly believe it myself!
In this proof Cantor makes of course use of the same assumptions as in other proofs. His treatment of v0 is therefore wholly understandable within his framework. But the fact that he assumes that it does not apply to a is simply staggering. 
I would except a statement like this
i) a.w=w,
but instead I get
ii) a.w=aw

In other words, when dealing with numbers of the second number-class, Euclidean logic is magically restored!
[book IV of the Elements, "Theory of Proportion"]

How is that possible? I might have missed something of course, but I could not find any explanation in the preceding sections. Why the normal rules of arithmetic suddenly become valid again when we are dealing only with numbers of second number-class is something I cannot, as of now, explain.

The Liar Paradox (and other beasties)
Cantor's Logic (15.4) The Numbers of the Second Number-Class Z(aleph zero ) (end)
[Don't forget to keep the book close at hand! For those who can speak German I advise Zermelo's edition, but not in digital form. The different symbols tend to be blurred in pdf files.]

Proof of H
"H. If a is any number of the second number-class, then the totality {a'} of numbers a' of the first and second number-classes which are less than a form, in their order of magnitude, a well-ordered aggregate of type a." (p.165)

This sounds like an other obvious statement, but we will see that it has far-reaching consequences that may, if not solve, at least throw some light on the mystery mentioned in the previous entry: how can transfinite numbers grow?

Proof.'Let F be a well-ordered aggregate such that F = a, and let f1 be the lowest element of F. If a is any ordinal number which is less than a, then, by section 14, there is a definite segment A' of F such that
and inversely every segment A' determines by its type A' = a' a number a' <  a of the first or second number-class. For, since Fc = ZO, A'c can only be either a finite cardinal number or ZO . The segment A' is determined by an element f' >- f1 of F, and inversely every element f1  >-  f1 of F determines a segment A' of F. If f' and f" are two elements of F which follow f1 in rank, A' and A" are the segments of F determined by them, a' and a" are their ordinal types, and, say f' -<  f", then, by section 13, A' <  A" and consequently a'  <  a". If, then, we put F = (f1, F'), we obtain, when we make the element f' of F' correspond to the element a' of {a'}, an imaging of these two aggregates. Thus we have 
But F'o = a-1, and, by theorem E, a-1= a. Consequently 
Since ao = Z0 , we also have {a'}c = Z0; thus we have the theorems :"

Let us stop here for a moment, and try to put all that has been said in this proof in a schema.

F'o=a; f1 lowest element of F;
________________________________ F, F'
f1   f'  f"          A'  A"

a' < a
a' < a"
f' corresponds to a': {a'}o=F'o.

First, a is presented as an ordinal number. Then it becomes an element. This metamorphosis has nothing of the innocence of Kafka's unfortunate hero. It allows Cantor to sneak in an illegal move and justify his conclusion.
The fact that it is formally valid is of course the whole point of the deception [this is again not a personal judgment, but the way the argument functions]. There is after all nothing wrong with turning a series of ordinal numbers into a set, and then comparing its properties to those of other sets.
Though it remains very suspicious when the ordinal numbers are those of the set F, the very set they are compared to. It allows the trick of declaring a'=a even if a' was first the number (the equivalent of a name here) of one of the elements of F, which still has a distinct element a! 
The ordinal number of the segment A' of F, had been declared to be a'. To now say that a'=a, which is the ordinal number of F, is to say that A'=F, which is an absurdity, or A'=F', which is just as bad.

Anyway, Cantor believes at the end of this part of his argumentation that he has proven that the cardinal number of the set of elements a' is the transfinite cardinal number aleph zero. He formulates this conclusion in a new theorem I, and add a theorem K, of still unknown taste, to his recipe

"I. The aggregate {a'} of numbers a' of the first and second number-classes which are smaller than a number a of the second number-class has the cardinal number ZO.
K. Every number a of the second number-class is either such that (a) it arises out of the next smaller number a-1 by the addition of 1 :
a=a-1 +1,
or (b) there is a fundamental series {av} of numbers of the first or second number-class such that
a = Limv av." (p.166-167)

We have now to wonder whether those theorems can be considered as logical consequences, or whether they still deserve their own proof, as Cantor thinks they do. He does not say what the following proof is supposed to be a proof of, only of K, or also of I. But it soon turns out to be the proof of K alone. Which of course makes sense, since theorem I was just the reformulation of the conclusion of the previous proof.
Let us first make sure we understand what theorem K is about.
It concerns obviously the way numbers of the second number-class are generated, or born.

[Who cares really? If you want to be a Platonist, be my guest!
[George: Eray Ozkural does.
me (waiving my hands up and down and whispering): shhht! He might hear you.
George (whispering back): Eray Ozkural does.]

Condition (a) sounds quite universal and therefore uncontroversial. So, it all comes down to condition (b). The limit of a fundamental series, whether of the first or the second number-class, can be considered as the next higher number. Quite a responsible job for such an elusive employee that never seems to be home!

Proof (p.167-169? It is unclear where it stops). 
i) "Let a = Fo. If F has an element g which is highest in rank, we have F = (A, g), where A is the segment of F which is determined by g. We have then the first case, namely,
a=Ao+1=a-1 +1
There exists, therefore, a next smaller number which is that called a1."

As usual, Cantor creates an extra element, segment A of F. We are obviously treating of the first condition, where a comes after a-1. Cantor labels it a1.
It remains obscure though why the fact that F has or has not a highest element should play any role so far. Apparently it is no problem if it does have a highest element, (ii) treating of the other case:

ii) "But if F has no highest element, consider the totality {a'} of numbers of the first and second number-classes which are smaller than a. By theorem H, the aggregate {a'}, arranged in order of magnitude, is similar to the aggregate F ; among the numbers a', consequently, none is greatest."

The identity of the {a'} and F is, as I have hopefully shown, certainly problematic. But first, it has to be made clear why Cantor needs this identity. It appears that in this case it does not matter at all that {a'} is declared equal to F, for the simple reason that we are not dealing with how F is formed, but with the question whether a' should be considered as less than Limv av, which will be always the case.
However, why should it matter that "among the numbers a', consequently, none is greatest"? Cantor is, according to the formulation of theorem K, supposed to prove how each time the next higher number is created, not necessarily the highest number.
Let us keep this in mind for the moment. 

iii) "By theorem I, the aggregate {a'} can be brought into the form {a'v} of a simply infinite series. If we set out from a'1, the next following elements a'2, a'3, . in this order, which is different from the order of magnitude, will, in general, be smaller than a'1; but in every case, in the further course of the process, terms will occur which are greater than a'1; for a'1 cannot be greater than all other terms, because among the numbers {a'v} there is no greatest." (p.168)

We are here confronted with a piece of Cantorian rhetoric we have encountered before. "simply infinite series" seems here to be a euphemism for what I have called a "multi-dimensional object" in Cantor's Logic (9.0). You never know in which direction to look for the next highest or the next lowest number in the set (remember, there is no absolutely highest). But somewhere in this imploded universe, "terms will occur which are greater than a'1". And just as in section 9, we are supposed to go along with the singular method which consists in choosing each time the number with the lowest index, and which we magically seem able to distinguish from all others in those infinite dimensions.
We understand now why it mattered to Cantor that no element was the highest. He needed chaos to be real, before he could impose his own order. Which he does in the long paragraph that follows (and which I will not analyze in detail).
It is to be sure an artificial order based on the opportunistic assumption that (actual) infinity is not well-ordered, unless Cantor needs it to be. Cantor has not proven that numbers, be they natural, cardinal or ordinal, become members of a disorganized set [should I say sect?] when they reach infinity. Such an assumption is far from obvious. Still, he, but mostly his followers, will spend a lot of energy in trying to prove the necessity of the so-called Well-Ordering Principle, never standing still by the question whether this does not amount to an even bigger and more implausible assumption: that numbers are not well-ordered.

iv) "From the theorems B, C, . . ., K it is evident that the numbers of the second number-class result from smaller numbers in two ways. Some numbers, which we call "numbers of the first kind (Art)," are
got from a next smaller number a-1 by addition of 1 according to the formula
a=a-1 +1;
The other numbers, which we call "numbers of the second kind," are such that for any one of them there is not a next smaller number a-1, but they arise from fundamental series {av} as limiting numbers according to the formula
a=Limv av.
Here a is the number which follows next in order of magnitude to all the numbers av
We call these two ways in which greater numbers proceed out of smaller ones "the first and the second principle of generation of numbers of the second number-class." (p.169)

The tone of (iv) could be said to triumphant in that Cantor is convinced he has shown how numbers of the first and those of the second number-class are created. But his victory is built on the illusion that he has succeeded in distinguishing between finite numbers on one hand, and infinite numbers on the other. 
But what has he accomplished really?
The fact that there is never an ultimate highest number does not automatically make numbers which depend on this limit for their existence different from those which do not. And nothing in Cantor's argumentation shows any distinctive property by which we could separate one group from the other. Even if we concede that Cantor has proven that some numbers are created by considerations of Limits, while others can simply be created by adding 1 to an already existing number, Cantor has not proven that they cannot be the same. That a number only hinted at by a limit cannot become 
a-1 +1.
Also, and I consider this more fundamental, the a in Limv a, is not (just) a number. It is part of a procedure whose meaning is to create numbers, or at least, the possibility of numbers.

The Liar Paradox (and other beasties)
Cantor's Logic (10) The Fundamental Series contained in a Transfinite Ordered Aggregate

I found the following points worth further analysis in sections 10 and 11.

The Imploded Number Line
"An ascending fundamental series {av} and a descending one {bv} are said to be "coherent," in signs
(5){av} || {bv},
if (a) for all values of v and u, 
av -< bu,
and (b) in M exists at most one (thus either only one or none at all) element m0 such that, for all v's,
av -< m0 -< bv." (p.130)

I find this a very strange affirmation. Take the infinite series containing negative and positive integers. Any point can be considered as the origin, and will therefore follow all those less than itself, and precede all those higher than itself. So, what does Cantor mean, if not that?
A less obvious use of the principle would mean that there can be no two origins that are distinct and equal at the same time.
We can choose any origin we want, but no other origin will give the same result. Not in the same series at least.

This is the birth of the "imploded universe" or "multi-dimensional objects" I mentioned earlier (9 and 15.4). Note that Cantor plays on the one dimensionality of the fundamental series on one hand, and the multi-dimensional character of the set M containing all those series on the other hand. That allows him to switch his perspective without explicitly declaring it, like I showed earlier. [See also (*) below.]

"B. Two fundamental series proceeding in the same direction of which one is part of the other are coherent.
If there exists in M an element m which has such a position with respect to the ascending fundamental series {av} that :
(a) for every v
av -< mO ,
(b) for every element m of M that precedes m0 there exists a certain number v0 such that
av >- m, for v >= v0" (p.130-131)

It should look like this:


"then we will call m0 a "limiting element (Grenzelement) of {av} in M " and also a "principal element (Hauptelemenf) of M." In the same way we call m0 a "principal element of M" and also "limiting element of {bv} in M" if these conditions are satisfied :
(a) for every v
bv >- m0,
(b) for every element m of M that follows m0 exists a certain number v0 such that
bv >- m, for every v >= v0."


The same variable v is used for elements preceding and elements following m0, which is a little bit confusing.
This concerns obviously the one-dimensional series, what Cantor calls the "fundamental series".

The following statement is in fact what the whole is all about:

(*) "A fundamental series can never have more than one limiting element in M ; but M has, in general, many principal elements."

What remains astounding is how such a set, which reminds me of the magical bucket our djinn used, can even be possible. The assumption of actual infinity alone makes it conceivable: all possible infinite series together in one bucket, or set.
Unlike the djinn with his bucket out of which the magic creature could only pull one single digit at a time, Cantor is able to pull any order he wants out of the set, or point at its inherent anarchy when needed.
Cantor never speaks (starting at section 9 ) of type eta other than a one dimensional object like the number line. The property of "everywhere dense" is also understood as local black holes, as it were. Wherever you stop you risk drowning in the infinite depths of each point. This way, the idea that any element has an infinity of elements preceding it keeps its one-dimensionality.
But it is a fake one-dimensionality. You enter each time an infinite universe, which makes the number line as complex as an infinite cluster of galaxies. 
Chaos is luckily the savior of rationality. Because there are apparently no rules of precedence, the mind is free to impose its own order. Otherwise it would go from one preceding element to the next, never sure if it is still on the same series, or whether it has inadvertently switched dimensions. The mind would for ever be trapped in the same diagonal maze Cantor had devised. A rational trap, that's for sure, and for that very reason, much more dangerous than chaos which can be sculpted at will.
Cantor never explains to us how it is that we are able to solve this modern version of Zeno's paradox. How come we go so easily from one number to the other while counting for instance? Apparently, the mathematician's freedom is in itself enough to keep him from drowning in all those little black holes. You have to choose to enter the diagonal maze, you cannot just tumble into it, like Alice does in the rabbit hole.
It is also important to realize that the rabbit hole does not explain the breakdown of logic. Infinity does not necessarily mean that n+n=n, or n.n=n. There is no inherent logical necessity between both ideas. Nothing logical prevents us from going down the infinite series of irrational numbers one number at a time, and in any direction. Each step would be in itself a logical step, easily explainable by its history. That our mind would eventually implode is a human idiosyncrasy, not the end of logic.
And that is exactly what Cantor wants from us, to give up our belief that 1+1=2 is valid even in the rabbit hole.
Actually, Cantor only wants us to suspend that belief, not give it up entirely. Otherwise we could never get from one level of infinity to the other. "1+1=2" needs sometimes to be true. The whole point is to convince us that there is a logical pattern to the on and off switching of our very logical intuitions.
I am afraid, I am still unconvinced.

The Liar Paradox (and other beasties)
Cantor's Logic (11) The Ordinal Type theta of the Linear Continuum X

[I will use T for theta]

The Continuum or Metaphysics 101 for Mathematicians
The idea of the Continuum has kept many great minds busy. Its tantalizing promise of a final, mathematical solution to the problem of infinity was irresistible for many. You must of course be able to understand and create complex mathematical formulas that look like magical pentagrams to the non-initiated. It gave mathematicians the unique chance to discuss metaphysics without leaving their numbers at work. How could they ever refuse?
My aim is much more modest, I am still sticking to the text and trying to determine what Cantor has achieved in the Contributions. The all-encompassing discussion will, I am afraid, have to wait.

The X-files
First two important definitions from the previous section:
"If an aggregate M consists of principal elements, so that every one of its elements is a principal element, we 'call it an " aggregate which is dense in itself (insichdichte Menge) If to every fundamental series In M there is a limiting element in M, we call M a "closed (abgeschlossene) aggregate." An aggregate which is both "dense in itself" and "closed" is called a "perfect aggregate." (p.132)

Cantor starts this section with a seemingly very clear proposition:
"We turn to the investigation of the ordinal type of the aggregate X={x} of all real numbers x...". (p.133)
But then, he shatters our relief a little bit later, declaring:
"The aggregate X contains as part the aggregate R [of rational numbers] of ordinal type eta" (p.134).

A set is therefore said of type T if it is perfect, and if between every two reals, there is a series of rational numbers. Cantor then presents the following corresponding theorem:
"If an ordered aggregate M is such that (a) it is "perfect," and (b) in it is contained an aggregate S with the cardinal number Sc=Z0 and which bears such a relation to M that, between any two elements m0 and m1 of M elements of S lie, then Mo = T." (p.134)

I almost skipped the proof which sounded like so many others in the Contributions, when my eye fell on the following statement:
"since S is a part of M, between any two elements s0 and s1 of S other elements of S must, by (b) lie."
And I thought to myself: what a wonderful world, the number line has gone nova!

The rest of the proof is metaphysically totally uninteresting.
[George: did you just say "totally"?
me: yes, why?
George: nothing. I just thought that only teenage girls used that word.
me: mathematicians speak of "totally ordered sets".
George (chuckles): I wonder what parents would think if they heard their daughter say that to her BFF on the phone. They would probably check if they still had their credit cards!]

Anyway, this is what Cantor had promised us:
"We will now show that these properties, taken together, characterize the ordinal type T of the linear continuum X in an exhaustive manner". (p.134)
And this is what he concludes:
"The proof of the similarity of X and M is now finished, and we thus have Mo s~ T" (p.136)

Rather disappointing, is it not? He defines first these properties, then he shows that they exist between M and S, just like they exist between X and R. He could repeat the same proofs for an infinite number of other sets, that would still no tell us anything we did not know when we first read the description he gave of X and theta. And that is that the Continuum consists of real numbers series interspersed with rational numbers series. And that it is, of course, infinite.
No wonder Cantor turned to theology at the end of his career, formulas cannot replace prayers when you seek contact with the Almighty.

For those interested in the proof, I have the following drawings to show the logical relations between the different parts. You have, somehow, to imagine all the parts fitted together. I am afraid that is the best that I could do. You will also need the text to go along with it.

_________r___________ R

_________s___________ S
                  s            S m0
_______________________________ M
                  r            R x0 [does not belong to R]
_______________________________ X
                  xv [fundamental series with x0 as limiting element]


The Liar Paradox (and other beasties)
Cantor's Logic (16.0) The Power of the Second Number-Class is equal to the Second Greatest Transfinite Cardinal Number Aleph-One

The most interesting part in this section is heralded by this declaration:
"We will now show that the power of the second number-class is different from that of the first, which is ZO." (p.171)

Proof of D
[d is delta]
"D. The power of the totality {a} of all numbers a of the second number-class is not equal to ZO .
i) If {a}o were equal to ZO, we could bring the totality {a} into the form of a simply infinite series y1,y2,..., yv, . ..
ii) such that {yv} would represent the totality of numbers of the second number-class in an order which is different from the order of magnitude, and 
iii) {yv} would contain, like {a}, no greatest number.
iv) Starting from y1 let yp2 be the term of the series which has the least index of those greater than y1, yp2 the term which has the least index of those greater than yp2 , and so on. 
v) We get an infinite series of increasing numbers,
y1,yp2,...,ypv,..., such that
1 < p2 < p3,... < pv < pv+1 < . . .,
y1 < yp2 < yp3... < ypv < ypv+1...,
yv =< ypv.
vi) By theorem C of section 15, there would be a definite number d of the second number-class, namely, d=Limv ypv,
which is greater than all numbers ypv
vii) Consequently we would have d > yv for every v. 
viii) But {yv} contains all numbers of the second number-class, and consequently also the number d ; 
ix) thus we would have, for a definite v0, d=yv0,
x) which equation is inconsistent with the relation d > yvo
xi) The supposition {a}o = ZO consequently leads to a contradiction." (p.171-172)

The first statement of the reductio, (what else could it be?) is what must be proven false. This a crucial step, so we must look at it very closely.
Cantor is convinced he can ask of us to believe that we can decide when an infinite series is equal to aleph zero. He makes it sound like a definite number, but we know that nothing in his analysis, even assuming it was all correct until now, can sustain such an assumption. Let us look at aleph zero more closely, and review the results obtained in Cantor's Logic (6.0) and following.
I concluded 6.0 with the following statement: "That brings [us] to my last remark, for now, on Aleph Zero. It is, as we know, defined as the totality of finite cardinal numbers. Not as the finite totality of finite cardinal numbers. Finite numbers are infinite in their own right. Let us not forget that."
Aleph zero might be defined as the cardinal number of the set of all finite numbers, but it is certainly not a definite number itself.
That might be one of the reasons that so many great scientists had no trouble with arithmetical operations having not the normal and expected results. Saying that Z0.Z0=Z0 is certainly not the same as saying that 23.23=23, or whatever large number you might imagine.
The concept of aleph zero seems to be the embodiment of the mathematician's freedom so dear to Cantor.
The distinction between aleph zero, the cardinal number of all finite numbers, and v, the set of all finite numbers, is also a very smart move. v can be understood in two ways:
- as the name of the infinite series of finite numbers,
- as the highest number in this series at any given time. As such, its value is always temporary, and it can only get bigger just by looking at it. [don't say it, George.]
Aleph zero is, in some way, v as it should be. Freed from the pesky necessity of growing compulsively. Aleph zero is v in perpetual rest.
As such it inherits the property of being a definite number without the costs of dynamic infinity.

This way, Cantor has the best of both worlds: he can use aleph zero as the undefined quantity of all finite numbers, but also as a definite number which gives passage to higher levels.
Notice that in its former shape, aleph zero is indistinguishable from v. Just like the latter, we could say that v+1=v, or even that v to the vth power is equal to v. But while v is forever stuck in its epileptic convulsions, aleph zero can take the shape of a definite number that gives birth to its heir, aleph one. How it does that could easily be compared with the dropping of a new baby in mother's lap by the stork when other children are not looking.
We know now that aleph zero, just like the good fairy, does not act directly but by proxy. It engages the services of a venerable mathematical figure, the limit of v, which is then endowed with the same magical properties aleph zero had received from its creator: its double nature as an ever growing entity, and a definite number with a precise value.
Without this precise, even if never explicitly stated value, the corresponding affirmation "If {a}o were equal to ZO" in (i) would be meaningless.

Let us note the curious remark in (ii), "in an order which is different from the order of magnitude". I must confess that I have no idea where this comes from [unless we consider the implosion mentioned below], but it does create its own difficulties. How do you know, even assuming actual infinity and an infinite time to study the series and its elements, that you have had them all if they are not in some order of magnitude, whatever it may be?

While (iii) reminds us that we are dealing with infinite series, (iv) throws us in the whirlwind of the imploded number line. Each element of the series branches in an infinite number of other series, which luckily, as always, contain in them enough identifying signs to choose the one with the least index among them.

This allows us to skip everything from there until the most surprising assertion of (viii):
"But {yv} contains all numbers of the second number-class, and consequently also the number d". In other words, a set contains its own limit!
If that were the case, how could aleph zero ever have been born out of the set of finite cardinal numbers? It would have been, just like the limitv v, merely another member of {v}! We must not forget that limv v is equal to the ordinal number omega, and the cardinal number aleph zero (section 15 p.160).
This would mean that we could use this proof as an argument that the existence of aleph zero leads to a contradiction! All we would need to do is change "second number-class" into "first number-class".

Aleph One, just like Aleph Zero, is based on very shaky logical grounds.

The Liar Paradox (and other beasties)
Cantor's Logic (16.1) The Power of the Second Number-Class is equal to the Second Greatest Transfinite Cardinal Number Aleph-One (end)

Proof of E
"E. Any totality {b} of different- numbers b of the second number-class has, if it is infinite, either the cardinal number ZO or the cardinal number {a}c of the second number-class."

As the proof shows, this is a variation on Proof of C in 15.2. 
Move along folks, nothing to see.

Proof of F
"F. The power of the second number-class {a} is the second greatest transfinite cardinal number Aleph-one." (p.173)

I sincerely hope this one will be more interesting.

i) There is no cardinal number a which is greater than ZO and less than {a}c
ii) For otherwise [the English translation says: for if not", which sounds very clumsy to my ears], there would have to be, by section 2, an infinite part {b} of {a} such that{b}c = a. 
iii) But by the theorem E just proved, the part {b}c has either the cardinal number ZO or the cardinal number {a}c
iv) Thus the cardinal number {a}c is necessarily the cardinal number which immediately follows Z0 in magnitude; we call this new cardinal number Z1. {Aleph One]
v) In the second number-class Z(ZO) we possess, consequently, the natural representative for the second greatest transfinite cardinal number Aleph One." (p.173)

(i) is what has to be proven. No reductio this time. We have the second number-class {a}, whose cardinal number is a, and there is nothing between it and the preceding cardinal number Aleph Zero.
Which means that we can consider them being in the same relation as 0 and 1, or 1 and 2.
This makes sense somehow, since anything we could add to aleph zero would leave it unchanged. We cannot therefore get from one aleph to the other by successively adding 1 to it, or even multiplying it by itself. How can we do it then? How do we go from aleph zero to aleph one? I am afraid Cantor is not ready yet to disclose his secret, and I wonder if he ever will. The rest of the proof is as disappointing as the previous one, or the one in 15.2.
In short, a cannot be aleph zero, so it must be aleph one. 
So much for faith.

This is the end, my beautiful friend
The rest of the article might as well be a text of the kabbalah as far as I am concerned. It is mathematical speak for mathematicians, and I gladly leave its evaluation to them.
I would be very surprised if Cantor somehow succeeded in solving all the defects of his systems in the following pages. Again, that is for mathematicians to decide.
As a philosopher I can sincerely say that I see no reason to take his views on infinity seriously. He could not justify the abandonment of logic by the creation of his aleph's. His proofs were generally very weak, when not outright inexistent, relying more on rhetoric and obscurity than on logic.
The fact that I am not a mathematician may have worked in my favor. I was not easily distracted by the apparent formal validity of his arguments. In a way, I was like a child annoyingly nagging the adults with his why questions. What maybe seemed obvious to mathematicians was for me very often quite problematic.
Also, like I said before, Infinity is a fascinating topic not only for philosophers but also for mathematicians. This latter group is known for its rigor and no-non-sense attitude when it comes to proofs. But they are also as human as any other people, and the desire to solve such an intriguing puzzle is of course present in each one of them. Not being philosophers, they were probably more concerned with the mathematical aspect of Cantor's presentation, than with its metaphysical significance. But Infinity is a metaphysical problem par excellence, and the idea that formulas can solve metaphysical problems is not a mathematical principle. It is a metaphysical conviction. With Gauss, I would urge the mathematicians to reconsider the (philosophical) value of Cantor's system, and look at Infinity as indeed a mere façon de parler.

The Liar Paradox (and other beasties)
Dear El Ouggouti,
 How much does positivity influence good natural satisfying philosophy? That is every discipline is a training for the mind and solely this purpose. Now the greatest discipline is philosophy. Whether it is practical, taught, part of everyday life etc. Not ocean is deep enough that it has no bottom just as all water has a surface. No ocean is fathomless therefore every philosophical question is answerable. Sometimes with more than one answer. Infinity exists only in the heavens. For that reason an attribute of, for those religious philosophers, divinity. I would now like to look at Pythagoras. No mathematical truth is empirical. Again infinity, an attribute of divinity proven using creation. Power of a divine is creation, sustenance and dissolution all at once with power remaining unchanged. But is power or the forces of nature a rational truth like mathematics? I would say yes but is dependent on time therefor falling short of the perfection of mathematical truths. Mathematical truths are beyond our world but equally of significance is power which gives rise change in the natural world. I could try to put this in mathematical terms and therefore extract the truth, but I do not have time and may leave for later stage. Anyone else is welcome to attempt. But to my knowledge because of the depth of thought, and thinking is as individual as a fingerprint, in equating, it is the configure who is best capable.

I just need to add, nectarines have a factor in likeness, because they have a family. Such is the construed 'nature' of mathematics.