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Set Theory: Mathematics or Metaphysics?
Preliminary Observation
Set Theory is believed to be the foundation of Mathematics, the theory from which everything mathematical would be deduced. It sounds like a metaphysical prejudice to my ears: how could a mathematical theory ever found mathematics? What would then found Set Theory itself? Its axioms? They are all of a mathematical nature, so that would not work. We would need non-mathematical axioms to found Set Theory, before it ever could found Mathematics. Is that even possible? This is what I intend to research in this thread. But please, bear with me, there is no royal road to the foundation of the foundation of Mathematics. If there even exists such a thing.

Axiom of Choice and Well-Ordering Principle
The universal consensus is that WOP relies on AC. I have the strong impression that it is in fact the other way around.
When I look at the incredibly complicated "proof that every set can be well-ordered" by Zermelo (1904), I cannot escape the feeling that he would not be able to choose one element from each subset, let alone, point at a distinguished element of a segment, if he could not distinguish those elements as preceding or following each other. And isn't that exactly what is meant by a well-ordering principle? Imagine that the sets Zermelo was dealing with did not contain numbers but any other kind of objects. Let us say tennis balls. How could he ever distinguish one group from the other, or name a segment as being a part of a set A but not of a set B? Even if we made it easier for him by making each ball have its own color, or even easier, each group of balls with its own color! 
Mathematicians like to emphasize that Set Theory does not concern numbers specifically, that its rules are valid whatever the nature of the sets involved, but who are they really kidding, if not themselves? Set theory without numbers would be a joke not worth telling twice.

Set Theory: Mathematics or Metaphysics?
The idea that formulas can only be interpreted in one way keeps the Logic industry in business. After all, that is one of the most important arguments in favor of the so-called mathematization of Logic and the severing of the umbilical cord with the biological mother: philosophy.
I think that was the greatest mistake European university administrators ever made (do not mistake the decision for a pure technical matter, a lot of prestige was involved at both sides). It encourages the myth that syntax is not only neutral, it is the only way to the truth. It makes also possible the creation of metaphysical monsters without anyone being aware of it for the simple reason that the only group who understands the language is too busy creating them. The adoration of formulas opened the way for Cantor to dominate the metaphysical scene in mathematics for more than a century without anyone daring close the doors of his alleged paradise. It is what still makes it possible for mediocre mathematicians to claim the title of neuro-scientists for the absurdities they publish in very expensive books.
That a formal language is no guarantee of infallibility may be illustrated by this small, some will say petty, example from Raymond Smullyan "Set Theory and the Continuum Problem", 2010 :

"Definition 2.1 
A class A is called transitive (sometimes complete) if every element of A is itself a class of elements of A-in other words, every element of A is a subclass of A." 

This is the formal translation:
0) "(Vx)(Vy)[(x E Y /\ YEA) => x E A]". (p.16)

Let us look at it in detail:
i) (Vx)(Vy)
For every x and every y
ii) (x E y /\ yEA) 
if x belongs to y AND y belong to A,
iii) => x E A 
THEN x belongs to A.

How about we simplify (0)?
0') (Vx)(Vy)[(x E y /\ yEA)]
x belongs to y AND y belong to A.
We would be left with
iii) => x E A
But doe we really need (iii)?

There is also the matter of sub-class and elements. Apparently, y is not only, just like x, an element of A, it also a sub-class. Can we read that off the formula?
"x belongs to y AND y belongs to A", (ii) is crystal clear. But is the conclusion (iii) true, that x belongs to A?
you can only be a child of y if y is a parent. That does not make you a parent, does it?

In fact, Smullyan is here stating the conditions which make a set transitive. In other words, the conclusion is not (iii), but "A is transitive". This is, more or less, what the argumentation is supposed to look like:

1) IF x belongs to y
2) AND IF y belongs to A
3) AND IF [we can say] "THEN x belongs to A"
4) THEN A is transitive.

Again, you might wonder if we really need the extra step(3).
Anyway, translating all four  statements we should have something like

(0") A(T) {(Vx)(Vy)[(x E Y /\ YEA) => x E A]}

Still, even the computer knew something was not right. This the result of copy and paste of (0)

(Vx)(Vy)[(x E Y /\ YEA) :) x E A]. 

Who says computers have no sense of humor?

Set Theory: Mathematics or Metaphysics?
What are sets made of?
That numbers are ubiquitous in Set Theory will be apparent from this example taken from Felix Hausdorff "Set Theory", 1924/2013:
"Let A = {5, 6, 7, . ..} be the set of all the natural numbers from five on, В = {1, 2, 3, ...} the set of all the natural numbers, and C={1, 2, 3, 4, 5,6} the set of the first six natural numbers. 
Then we have В - A = {1, 2, 3, 4}, С - (В - A) = {5, 6}." (p.15)
First, it would seem that we could easily replace those numbers with variables that could designate any set of objects. Let us do it.

i)   A={a,b,c,...},
ii)  B={d,e,f,..,}, 
iii) C={a,b,c,d,e,f}
iv)  B-A={a,b,c}
v)   C-(B-A)={e,f}.

Still, those elements need to have at least those two properties:
1) They must be ordered, otherwise we would not know how to subtract one set from the other (unless the sets are finite and easily overseen).
2) each element is distinct (and distinguishable) from the other.

If those elements were tennis balls, we would need a distinct color for each ball. Also, we would need a rule that tells us which color precedes or follows each other color, and that for all colors. That is the only way we can be allowed to use open expressions like (i) and (ii).
In other words, we may be able to work out this kind of examples when dealing with small finite sets, but it would soon become practically impossible to do that without numbers.
Those color schemes are therefore nothing else but a, very impractical, numeral system. As is a series like


Mathematicians may prefer to think that Set Theory is the foundation of Mathematics because it explains numbers at a more primitive level, a closer analysis shows that its concepts are at least as sophisticated as the rules of arithmetics in particular, and Mathematics in general. And that it would be very difficult to distinguish between them.

Set Theory: Mathematics or Metaphysics?
The Well-Ordering Principle and the Djinn

[George: shouldn't he get a name? I find it so insulting to refer to him by his race. You might as well call him "nigger" or "honk".
me: you think so?
George: Absolutely! After all, you don't me call me "hey! Homunculus!". I say we call him ... Shaito san. You know, from Shaitan, plus a Japanese tint for universality?
me: Shaito san. Sure, sounds good to me.]

So, Shaito san [I must admit, it asks some getting used to] starts shaking his magical bucket, then grabs an infinite bunch of numbers out of it, before splashing them against his wall [well, he must have a home somewhere, don't he?
[George: no Sanny, you are not allowed to look! (to me). Otherwise it would be no fun. He can see the whole set in a glimpse!
me: Sanny? What happened to Shaito san?
George: ah well, we imaginary creatures are somewhat easy-going with each other. But you, you should really stick to Shaito san. And don't ever drop the san. That would be quite rude!]

With the precision brought in by George, that we are in fact faring blind, we can ask the question again. Can this random infinite set be well ordered?
Allow me first to speak only of numbers and leave in the middle the question of ordinals and cardinals. So all the elements on the wall are some kind of numbers, be they natural, algebraic, reals or imaginary, to name but a few.
Let us start with the obvious: there will be many duplicates, so I propose to ignore them, at least for the time being. Imagine you have a few decks of cards shuffled together and all you need is a poker set. How do you do that?
Well, you start of course by putting the first card aside, whatever it is. It will always be a valid poker card.
You then compare the second one with the first one, and if they are not equal, you put it also in the right corner. And so on.
Such a method would only work if

1) you are dealing with a small finite set,
2) you can immediately decide whether a card belongs to the right set or not.

What if both these conditions were absent?
A simple ordering system is the < (or >) relation. If a number x is < y than we put it to the left of y, otherwise to the right.
Also, if it is a complex formula, we will need first to find its solution, if it has any, before assigning it a place. By the way, in case it does not have any solution, we can always place it at the right anyway, and have it shift places any time we get new elements.
Which means of course that the number of unsolved formulas will get larger and larger.
George: where are you getting the formulas from?
me: what do you mean, where I am getting them from? They represent numbers, don't they?
George: San says he does not have any formulas in his bucket, only numbers.
me: you sure about that?
George: that's what he said. But you can always ask him yourself.
(Shaito san grunts.)
me: nah, it's okay, I believe you. But does that mean that he has only natural numbers?
George: no, just like you said, any kind of numbers, just no formulas.
me: why? Why no formula's?
George: because it is something that humans do. Instead of using numbers directly, taking them out of the bucket as it were, they have first to find them using formulas.]

Let us take Hamel "Basis aller Zahlen", (1905), one of the few positive reactions to Zermelo's first proof.
What he says boils down to the fact that without AC he would not be able to prove certain theorems. Which formula or theorem is not really important, the point is the structure of the argument:
By choosing arbitrary functions or relations relevant for his objective, he is able to build up a proof he considers as valid.
The question is of course, what justifies these choices? A possible answer is that they seem evident in the context in which they are made. But then, could there be a rule at the basis of these choices?
This is where it gets really interesting.
Zermelo thinks that by proving that any set can be well-ordered, he can skip the necessity of an explicit rule for the choices to be made. Well-ordering make choosing a random path quite trivial. It would be like trying to find your way in a modern foreign city with all its street signs, but also armed with an up to date map. Which mathematician would not be able to find his way in such a situation?
But what happens if all the street signs are in a foreign language and alphabet, and the map is also in a different foreign language and alphabet?

So, here you are, with one of Hamel's formula's in one hand, and the unreadable map in the other, standing somewhere in a foreign city.
The formula says:
"There exists a basis for all numbers, i.e. there is a set of numbers a,b,c.. such that every number can be represented in one and only one way in the form
(1) x=aa'+bb'+yc'+..." 
[the first letters are the Greek equivalents alpha, beta, gamma,..., the second ones are the English letters without the prime sign.]

When we ask can (1) be well-ordered, what are we asking exactly?
Can we find numbers which make the formula true? That should not be a problem, after all, Shaito san has made an infinite set available to us.
Or do we mean maybe that only the numbers that make the formula true should be in that set? But what would be the odds of that if we are really dealing with a random set?
Wouldn't it be more reasonable to say that we will only pick those numbers that make the formula true in the first place? But that is the whole point, right? How would we know how to make those choices?
Shaito san said that there were no formulas in his bucket, only numbers, and that we humans made the formula's. In other words, each time we make a formula, we create a special link between us and the numbers that belong to it, a sieve as it were. And we do that even when we have no idea what those numbers precisely are. Is that magic? It certainly is. If you accept the idea of a magical bucket out of which all numbers can be pulled, then you should have no difficulty with accepting the fact that just thinking of a formula brings the right numbers to life. All we have to do then, is find them. And the fact that we cannot blindly point at them, does not mean that we are unable to recognize them if we bump into them. [this idea is certainly not new, I will mention the author that beat me to it when I find the right quote again.]
Anyway, what seems obvious to me is that numbers, while they are in the magical bucket, are just amorph collections of indefinite elements. It is only when we pull them out that they reveal their true nature. Which means very simply put, that you will pull only random numbers if you have no idea what you are doing. The counterpart of this should give us some hope: each time we handle a formula, we create a sieve which allows us to build up sets that make sense to us.

But we are allowed to go even further than that: random numbers, once out of the bucket, are still numbers. That means, with the exclusion of duplicates, that there always will be a (temporary) least and a (temporary) largest number in the set. Can we prove that? Sure, all we need for that is to prove that numbers are numbers. But that might be a little bit tricky. That may be why Zermelo's proofs were met with so much resistance. He tried to prove the obvious, which he of course could not, and found himself facing two different groups: one who believed in the obvious but was not convinced by Zermelo's arguments; and another group who rejected the whole proof because they found it unconvincing, even if they themselves often made use of the same axiom they were attacking. [My conclusions are somewhat different from Moore's "Zermelo's Axiom of Choice", (1982/2013) but I do not think they are incompatible.]

There is also the matter of the origins of the formulas. All the authors seem to think that these formulas simply spring out of the mind of mathematicians already fully formed, and that their genetic makeup is therefore a mystery that needs to be solved. Which it very often certainly does. But at the same time let us not forget that even when it concerns a formula you have not invented or discovered yourself, your formation and experience as a mathematician excludes many possibilities beforehand, and allows you usually to delimit quite clearly the range of acceptable solutions. That is already a form of well-ordering, how provisional it may be.

By the way, the Platonist formulation of the whole issue is purely rhetorical, feel free to adapt it to your own beliefs.
Anyway, I still owe you an analysis of Zermelo's proofs.

Set Theory: Mathematics or Metaphysics?
Simply ordered?
Well, I am afraid that nothing in Set Theory can be considered as simple. Take the trivial example of intervals as presented by Huntington in his "The continuum and other types of serial order: with an introduction to Cantor's transfinite numbers", 1917.
i) "For example, the class of all the points on a line between A and B, arranged from A to B, has no first point, and no last point, 
ii) since if any point C of the class be chosen there will be points of the class between C and A and also between C and B. 
iii) If, however, we consider a new class, comprising all the points between A and B, and also the point A (or B, or both), arranged from A to B, then this new class will have a first element (or a last element, or both)." (p.13)
This is a textbook case and therefore never put in question. But are things that simple really?
First, we acknowledge the existence of two boundaries, A and B, and then we act like they do not exist. I wonder what keeps us from applying (ii) to (iii). Why can we pinpoint A and/or B in all the other three cases but not in the first? There is no reason why we could not choose not to do so if we wanted to, but the impression all mathematicians seem to share is that there is something objective to this distinction, which goes beyond the preferences of the moment.
It is of course understandable that if we involve Infinity (ii) must be considered as valid. 
But didn't Cantor convince everybody that the segment of a line is equal to the whole line, and is therefore infinite? 
In this case, how could we ever make the distinction between open and closed intervals? And, if we can ignore this difficulty, why could we not ignore the first one, that says that the boundaries will forever be out of reach? What is actual infinity good for if not for that?
In other words, the distinction between open and closed intervals is untenable within a Cantorian perspective.
Infinity, just like beauty, seems to be in the eye of the beholder. And if that is the case, can we really say that a segment is equal to the whole line? 
Or that reals are not well-ordered because every real is preceded or followed by an infinity of reals?
Like I said before (Cantor's Logic 10), there is no reason to believe that we cannot go from one real to the other. In other words, every real will be preceded (or followed) by a single real. That each step depends on our own perspective and assumptions does not make it any less objective. It would be like jumping over a line, or landing right on top of it. It all depends on how far your legs can reach.
Saying that an infinity of reals precede or follows each real is therefore quite trivial. Whenever I am standing on the street, or in my kitchen, the numbers of paths I can follow can also be said to be infinite.
Between binary 101 and 110, there is only one single step, not two, and certainly not an infinity. And this holds for each step between two reals. Standing on 101, I can either choose to go to 110, or dive deeper in the series to 1001, and keep diving. But each time, the choice itself will be quite simple: Up? Down? Right? Left?
What Cantor and his followers do is forget to mention this special version of the "axiom of choice", even when they make explicitly use of it as Huntington in the quote above: the first case is when we keep diving, while the other cases all involve the conscious choice of taking one more step in the same direction.
The consequences of this conflict (finite steps vs Infinity) are something worth studying very closely.
Not the least one of them is that, in case we could not afford the services of an infinite number of djinns, we still can say that reals are countable given any perspective. They only seem uncountable when we want to run through all levels at the same time. The choice lies in which level we want to be active. In that, I agree with (Cantor and) Zermelo, even if I consider their proofs as totally inadequate.

Set Theory: Mathematics or Metaphysics?

Zermelo's first proof of 1904

i) "(1) Let M be an arbitrary set of cardinality m,

ii) let m denote an arbitrary element of it,

iii) let M', of cardinality m', be a subset of M that contains at least one element m and may even contain all elements of M,

iv) and let M - M' be the subset" complementary" to M'.

v) Two subsets are regarded as distinct if one of them contains some element that does not occur in the other.

vi) Let the set of all subsets M' be denoted by M.

vii) (2) Imagine that with every subset M' there is associated an arbitrary element m'1 that occurs in M' itself:

viii) let m'1 be called the "distinguished" element of M'.

ix) This yields a "covering" y of the set M by certain elements of the set M.

x) The number of these coverings y is equal to the product [...] taken over all subsets M' and is therefore certainly different from 0. In what follows we take an arbitrary covering y and derive from it a definite well-ordering of the elements of M."

The statements until (x) can all be considered as preliminaries. It start to be interesting with

xi) "Definition: Let us apply the term "y-set" to any well-ordered set My that consists entirely of elements of M and has the following property: whenever a is an arbitrary element of My and A is the "associated" segment, which consists of the elements x of M such that x -< a, a is the distinguished element of M - A."

This is how I understand the definition in (xi):

The segment A, of elements x, is separated from the rest of M by the distinguished element a, which is therefore the first element of M-A.


______________a___________________ M


The following statement (xii) is not as immediately evident.

xii) "(4) There are y-sets included in M. Thus, for example, [the set containing just] m1 the distinguished element of M' when M' = M, is itself a y-set; so is the (ordered) set M2 = (m1,m2 ), where m2 is the distinguished element of M - m1'."

This is how I understand it:

When M=M', it will contain only one element, m1. Then we get a greater M2 which starts at m2. Just like a was the separating element between segment A and the rest of M, m2 is the last element of the segment containing m1, and announcing M2. This way we can divide M in different subsets, and each time we will have a defining, separating, distinguishing element to pass from one part to the other. Since we can make each part or subset as small or as large as we want, depending on the function or covering used, we will have organized, or ordered, M in a series of those elements.

That explains at the same time

xiii) "(5) Whenever M'y and M"y are any two distinct y-sets (associated, however, with the same covering y chosen once for all!), one of the two is identical with a segment of the other."

As shown, each distinguishing element divides the set in a preceding segment and a following y-set, which will itself become a segment in the next phase.The rest of the proof is used to show that all the subsets taken together are necessarily equal to M. I will skip this part and go back to what I consider as more central to the proof.

Michael Hallett in his "Introductory note to 1904 and 1908a" in the first volume of the Collected works of Zermelo, emphasizes the fact that Zermelo builds up the different subsets, and does not just assume arbitrary sets. I agree with his analysis with the following reservation: how does Zermelo know how to build up his different gamma-sets?

As I said before, this procedure would have been impossible with (elements of) sets other than numbers. Without the well-ordering inherent to numbers, Zermelo could not have found his way to any other meaningful ordering. It is the difference between the "natural order" of (natural) numbers, and an order that only a mathematician can discern in a set of random numbers. (see Cantor's Logic 14.1).

It shows, I think, that the so-called Axiom of Choice, is indeed, and as Cantor first declared it, something more akin to a "law of thought". A Mathematician cannot look at random numbers and not try to find, behind their "natural order", patterns and regularities. Without this talent Mathematics would not be possible at all. The illusion resides in the expectation that he could somehow find an order in a random set without making any arbitrary assumptions at one time or an other. That would only be possible if the Mathematician could, with a single glimpse, just like Shaito san, see all the possibilities before his eyes, and a clear path through them all.

Set Theory: Mathematics or Metaphysics?
Zermelo's first proof of 1904 (2)
How can you well-order an arbitrary set? That seems to be the question Zermelo is asking. Such a question can be understood in two ways, as
1) a heuristic problem: what can the mathematician do to-well-order a set, or, as
2) an ontological problem: what can the mathematician do to unveil the hidden order within a set?

Both concern the actions a mathematician has to undertake, and can be therefore understood as an epistemological issue. But while the first is centered on the mathematician and his methods, the second one aims at disclosing the true nature of an arbitrary set.

You can, as a mathematician, be convinced that sets are amorph entities that only get their meaning within a theoretical setting. In this perspective you would look at a data stream as an invitation to try out certain hypotheses and see if they pen out.
You can also start from the conviction that the data has already a meaning, and that it is your task to find it.
In practice, a mathematician, or a scientist in general, will oscillate between those two attitudes indifferently.

What makes Zermelo' approach unique is its unilateral character. He is a convinced adept of the second view, that sets possess an inherent order and that the mathematician has therefore the possibility of analyzing each arbitrary set in a standardized way.
Zermelo seems to suggest that the Axiom of Choice, as It would be called in the future, points to one or more arbitrary functions that we assume at the start of our study, and which help us analyze any arbitrary set. But how could you describe such an arbitrary method? After all, the choice of the function is essential, it guides every step the mathematician will take in his study. But when we see what Zermelo proposes, it turns out to be a very specific way of looking at an arbitrary set.
While in all the preliminary statements until (x), the talk is still about arbitrary y-coverings, starting from the definition in (xi) Zermelo suddenly shifts gears and becomes very specific. He speaks in (xi) of "x -< a", or an element of a set preceding another. There is nothing arbitrary about that. The same principle is confirmed in the following statements where he speaks of segments and distinguished elements. This is a very limited view of the orderings the elements of an arbitrary set can have. In fact, the only sets which would seem to possess such a feature are number sets in their "natural order". Which would simply mean that Zermelo's method would turn any set of numbers in a naturally ordered set.
This allows me to pinpoint what I consider the fundamental flaw in Zermelo's reasoning: he turns a heuristic matter, how to approach arbitrary sets, into an ontological issue: what are the fundamental properties of any set? In so doing he misses his own objective which is to prove that the Axiom of Choice is a methodical necessity. The Well-Ordering he advocates as being overall present does not justify AC in any way. If anything, it could encourage the mathematician in thinking that all he has to do is continue unveiling natural laws instead of coming up with arbitrary functions wrought out of his own imagination. It would seriously hamper the freedom so dear to Cantor to approach mathematical reality unhindered by any preconception.

I would also like to point at a very strange aspect of Zermelo's approach. It somehow seems like the mathematician is approaching an arbitrary set with his eyes half-closed. When he looks at an arbitrary set, he is able to tell which elements belong to it and which do not, and what are its subsets; but to find the order between the elements, the mathematician has to go through strange methodical hoops to find the distinguished elements of each subset, and the different segments so created. 
That is of course the problem with any arbitrary set. Once you take it as an example, it becomes a very definite set with very definite properties. At the same time, because it is supposed to be an arbitrary set, you are not allowed to anticipate its features, even though they were already on your mind when you took, or at least looked at the example. So Zermelo has to act like he has to find the distinguished elements and their segments. Even though, or so it seems to me, they are what make subsets possible in the first place.
Maybe a concrete example will make this clearer:
A= {2,37,9,1,5} is an arbitrary set I just typed in with no idea what the possible relations between its elements could be. Zermelo's method would allow me to choose the -< relation and use it to order the set which would become
Then what?
Well, I am afraid that this is all Zermelo has to offer. I would not know what the use could be of distinguished elements and segments in such a situation. Would it change anything if the set was much larger? Or maybe if I marked the numbers in a number line then the meaning of 'segment' and 'distinguished element' would suddenly become obvious? This last point is I think essential. Zermelo does not seem to be able to understand sets other than as segments of a number line. I wonder what that says over the status of the number line in Set Theory. But then, historically, Cantor started with the study of points, Set Theory was first Point Set Theory. Something to ponder at leisure another time.

Zermelo's method maybe, if we feel very charitable, makes sense if a set has been created by some formula, and possesses therefore already a hidden order only waiting to be discovered. A completely random set may not have any ordering beyond the natural order of its elements. The mathematician might be able to distinguish different subsets, each with its own formula, but fail to find one or more  all-encompassing rules to explain all elements. This fact is what distinguishes a heuristic problem from an ontological issue. The link that Zermelo sees between the Well-Ordering Principle and the Continuum would seem to indicate that what he had in mind were not really arbitrary sets, but numbers in general, and reals in particular, and that, in their natural order:
"There must at any rate exist at least one such well-ordering, and every set for which the totality of subsets, and so on, is meaningful may be regarded as well-ordered and its cardinality as an "aleph". It therefore follows that, for every transfinite cardinality, 
m = 2m = Z0m = m2 , and so forth; 
and any two sets are" comparable" ; that is, one of them can always be mapped one-to-one onto the other or one of its parts."
The first part is a clear declaration of faith in Cantor's creation. The last sentence is utterly trivial.

Well-Ordering does not justify the Axiom of Choice. The former should be considered a consequence of the latter. Whether there exists a hidden order in a set or not, the mathematician has to (re)create it, and he does that by choosing arbitrary functions that help him circle the solution ever more closely.
Zermelo would like us to believe that every set is well-ordered, and that there is therefore a universal method of approaching arbitrary sets. If that were the case mathematics would be very simple indeed.

Set Theory: Mathematics or Metaphysics?
AC is provably equivalent to the well ordering principle - they are the same thing. So WoP doesn't rely on AC (or at the very least, phrasing it that way is very misleading). And it is precisely AC which states that there is a choice function - a way of choosing an element from any non-empty set. 

Set Theory: Mathematics or Metaphysics?
Also re your first observation: the claim is about the primitives of mathematics.

Set Theory: Mathematics or Metaphysics?
Reply to Carlo Lori
I suppose it becomes something of a chicken and egg issue when looked at it very closely. Still, the idea that "there is a choice function - a way of choosing" only makes sense in Zermelo's analysis because sets are well-ordered, or can be. I certainly agree with him when it comes to reals, I do not consider them any less well-ordered then other numbers, be they in a finite or an infinite set. What I find far less acceptable is the generalization of this Cantorian conviction to any arbitrary set. Zermelo is trying to prove that Cantor is right concerning not only reals but also concerning the Omega's and Aleph's. That is what makes his argumentation so biased, and therefore so unconvincing in my eyes.

If you understand AC as meaning: a mathematician can always find relations between elements of a set, then I would say it is either very trivial or pertinently false. After all, there are still unsolved conjectures, not to mention that that would negate the sense of "arbitrary" in a very mysterious way.
Concerning the question of primitives: an axiom does not need to be "right" to be useful in an axiomatic system. Unless of course, just like Zermelo does, it pretends to express an evident truth which, I concede you that, cannot be proven. Then it becomes a legitimate object of discussion.

The Axiom of Choice is what I would call a metaphysical monster: it takes an intellectual activity of the mathematician and attempts to turn into it an objective property of sets.

Set Theory: Mathematics or Metaphysics?
The chicken and the egg: a story without end

One to one correspondence is universally seen as the criteria of equality of two sets. It is, as far as I can see, a procedure, not a judgment, and can therefore not be a criteria of equality, if there even can be such a criteria, because only its result can claim to such a role. Let us look at how it works.
There are two sets A and B, and I want to determine whether they are equal to or different from each other. One to one correspondence is, as far as we know, the only way to go.
What do we do when we make use of such a procedure? We take one element of A and put it in front of an element of B. Both elements do not need to be themselves equal. One can be a car, the other a driver, or, an example favored during the time of the First World War, one a soldier, the other a rifle.
What are we comparing then? The difficulty we find in answering this simple question is a clear indication of the circularity of our reasoning when we attempt to define one primitive concept, number, by another, one to one correspondence. It is because one to one correspondence is used as a procedure, an action, and not a concept, that the link between them is acceptable to our mind. But as soon as we try to define the meaning of this procedure, then we are irremediably thrown back to the number concept.
Putting one object in front of the other is itself an act of counting. It is also comparable to the act of uttering a sound like 'one' each time we put an object aside.
We all know that numerals are arbitrary sounds that can, and are, easily replaced when switching from one language to the other, or from one system to the other. But the fact that numerals exist and have meaning for us seems as mysterious as the ability to put one on one two different objects.
Or rather, what is mysterious is only the concept of number. Once we accept its reality for and in our mind, then the procedure and the utterance are not mysterious anymore. Both express, in their own way, the "concept of number" [taken in a non-Fregean sense. Whether "number" can be considered as a concept is a question which I find much too complicated to even start to contemplate].
We turn the ungraspable and ineffable into something very concrete, an action and or a sound.
As I said, we find ourselves caught in a vicious circle as soon as we try to explain our actions: they refer to our concept of number, and the same concept can only be explained by our actions.

Let us now go back to one to one correspondence.
We are putting two (different) objects in front of each other, and repeating this act as long as there are objects present on both sides, or in both sets.
Where does that leave the conceptions of sets as a more primitive notion? (Smullyan:"the natural numbers can be explained in terms of the more basic notion of set...")
Is the notion of set indeed more basic than that of number? Is a singleton more basic than the simple concept of one? A pair, ordered or unordered, more basic than 2? Could we explain the latter, numbers, by the former, sets?
Let's try it.
What can it possibly mean to think of a set, which both pioneers, Dedekind and Cantor considered as an act of thought or creation of the mind?
Historians like to give the example of one ancient bone, from the dawn of Man, usually believed to be the most ancient evidence of the counting ability of our ancestors. What is remarkable to this bone was that it would be very difficult for a modern observer to discover anything resembling a set on it. There are a number of individual marks which experts bring in relationship with the tides of the moon, or some other calendar event; which means that each marking stands in principle isolated from the others, while at the same time sharing some general property only known to our long gone ancestor. It looks to me more like one to one correspondence than set theory.
Does that mean that the notion of set is more recent than that of number? I would not know. We are here speculating over a phase in the history of Man of which historical evidence is nigh inexistent. It seems to me though that the discussion which came first is pretty uninteresting. It would in fact be quite difficult to distinguish the notion of unity from that of group at such a primal phase. Does the lion see the buffalo herd before he [sounds so much more natural to me than "it"] sees the individual buffalo? Does he ignore the individual pieces of grass and soil on which he is walking, even though he avoids the individual rock or tree? Was Primitive Man much more different than a lion? 
These are all fascinating questions which lead us all to the same conclusion: we really do not know which came first, the individual or the group. Especially when we consider that individuals are indistinguishable from each other as far as primitive animals are concerned. 
What seems certain to me is that sets are more like procedures and numerals than like the concept of number as such, and could therefore no more explain number than one to one correspondence or numerals do.

Of course, nobody claims historical primacy of sets on other forms related to numbers, but set-theorists do claim that sets can explain numbers, and for that, a historical view can maybe put things in the right perspective.

Set Theory: Mathematics or Metaphysics?
Cantor's theorem
how much value should we give to it? It seems, at first glance, undeniable: the cardinal number of the power set of any set, is much higher than the cardinal number of the original set.
A set A with 3 members, will have a power set of 8 members, or 2 to the third power. But what does it tell us really? What are we allowed to deduce from such an empirical fact? Because, let us not be deceived by the fact that we are dealing with so-called acts of thought. In reality there are no power sets, they only exist in mathematics and in our own imagination. You can never turn a group of three elements into a group of eight, not without breaking them into smaller pieces. So what are we really talking about? But within mathematical Reality, there also exists a distinction between empirical and logical facts. Or at least there should be.
So, let us first agree to pursue the argument at the same level. There is no point in bringing everyday reality into the equation. That would only turn the problem into a metaphysical conflict between Platonists and non-Platonists. Like young students can express so vividly: boring!
Staying therefore within the realm of mathematics, or, what comes down to the same, the mathematician's imagination, we accept the distinction between (the cardinal number of) a set and (that of ) a power set.
Cantor felt justified in asserting that this distinction proves that power sets of infinite sets have a greater cardinality than those sets. Which makes of course perfect sense. Why should the rule suddenly change for infinite sets? You Set? Me bigger, me Power Set!
The problem only becomes obvious when Cantor tries to justify on this rational ground his idea of different levels of infinity. Which also sounds quite reasonable at first: let us call the cardinal number of all finite numbers Omega, then it stands to reason that the cardinal number of the power set of Omega will be bigger, doesn't it?
Well yes, if you could point at the cardinal number of all finite numbers, and put a definite value on it, then the cardinal number of its power set would certainly be much bigger. But what if you cannot put a definite value on that number? What if there did not exist such a number? Remember, we are still within the realm of the mathematician's imagination, where numbers and sets are as real as anything else in the world.
That still does not mean that anything the mathematician can imagine immediately gets promoted to a real existing subject (object) of the Realm. That would mean that there is no difference in this realm between Reality and Dream.
Of course, one's Dream world can be another's Reality, but can we assume that for each Mathematician at least, there is a, more or less, clear distinction between Dream and Reality?
The question therefore is, is Cantor's world Dream or Reality? Is he allowed to consider the cardinality of all finite sets as a number? Of course he is, Kronecker tried to forbid it and much good it did to him! So let me rephrase it. Is Omega, not to mention the Aleph's, a logical part of the Cantorian Reality, something that all mathematicians have to accept if they accept Cantor's Reality? Or can it be considered as his personal Dream world, in which others may or may not follow him?
Let us take a step back and look again very closely at the point of contention: does the cardinal number of all finite numbers exist?
We then see immediately that such a question is far from innocent. It would be like a catholic priest from the time of the Inquisition, asking innocently: does God really exist? In his mind he is posing a legitimate problem that should be approached rationally with all the intellectual tools in our possession.
Am I exaggerating? Hardly I would think. Cantor's Dream has created whole new branches of mathematics that would be unthinkable without this simple assertion: there is a cardinal number of all finite numbers, and it is less than some other cardinal numbers. By putting its existence in doubt we are in fact telling all this good, and very smart people that all the time and effort they have put in their research is meaningless, that they might as well have been studying the composition of phlogiston!

Let us recapitulate the uncontroversial results of Cantorian Reality: we are allowed to say
1) power set cardinality is higher than set cardinality
2) for infinite sets the same rule holds: how ever large an infinite set may be, its power set will be larger.

This has been seen by Cantor as a sign of contradiction: a sign cannot be larger than the largest sign possible. The way out of the contradiction seemed evident: either refuse the reality of power sets, or consider the cardinal number of all finite numbers not as the largest number possible. It still may keep the undisputed title of champion of all finite numbers, but it cannot claim anymore being part of the Big League. This honor is reserved for higher cardinals, the different Aleph's. To be precise, starting from Aleph1, Aleph Zero being the cardinal numbers of all finite numbers.
The beauty of the Aleph concept is first the dissolution of the contradiction: we get to keep the power set of all finite numbers, for the simple reason that it does not matter anymore. Aleph0 is just like a black hole, whatever you throw at it it swallows without giving anything back: even raising it to the same power as itself puts no dent in its armor.
Until now there is no reason for dissidents or skeptics to really revolt. It is after all nothing else but the idea that Infinity is so large that whatever you do to it it cannot ever become anything else but what it already is.
This is where all, or at least most of Cantor's contemporaries had wished he had stopped. They would have been more than happy to celebrate his genius for the results that he had already obtained so far: The precise difference between a finite and an infinite set, and the distinguishing properties of the different sets of numbers and their relation to infinity.
That was not enough for Cantor who described this infinity that nobody had trouble with, as "uneigentlich", or as Infinity non-proper. He would show the world what real Infinity was!
This is the crucial moment where Cantor's Reality turns into Cantor's Dream, the crossing point at which each mathematician is asked to declare his allegiance to the new realm.
Let us be clear: nothing in the Cantorian Reality compels one to make a choice for the Dream connection. If you do not believe in the Reality of cardinal numbers greater that those of finite sets, then there is nothing in the Cantorian analysis that could make you change your mind. The arguments, as is always the case with religious or metaphysical manifests, can only convince true believers, or skeptics heavily biased towards the Cantorian point of view. Because we are dealing with what I would call, without any pejorative connotation, a mystical engagement, we should consider the arguments in Cantor's writings not as purely rational statements, but more as an invitation to enter his Dream Realm, and help turn it into a new Reality.
D'ailleurs, that is apparently the way that it is generally understood by most if not all of his followers. I find it striking that there is not a single proof to find of the well-foundedness of the Aleph's in all the books that I have read on the subject. They all pay some kind of lip service to rationality by presenting a short version of what is supposed to be a very complex argumentation, but very soon turn to assuring and reassuring the reader that his belief not only is expected, but that it would be entirely rational and "scientific". The message is clear: if you want to be a good mathematician, you have to believe in the words of Cantor.
This simplifies the problem greatly. We should not waste too much time in proving or disproving the dogmas on which the Dream Realm has been built, that would show a lack of respect for all the hard workers who spent years of their life trying to turn Dream into Reality.
And when we remember that we are "only" dealing with the mathematician's imagination, then the conflict seems suddenly to lose all its sharp edges.
Whether cardinal numbers higher than the cardinal number of finite numbers exist or not in Reality, it remains a fascinating subject to research for mathematicians who have already decided not to be beholden to everyday reality (with a small 'r'). It would be in fact the best tribute that a mathematician could present to Cantor and a confirmation of the spiritual freedom of the mathematician: if Reality throws obstacle in our path, we are free to imagine another Reality, one that starts like a simple Dream, until we turn it into a full-fledged Reality by the mere power of our minds.

Set Theory: Mathematics or Metaphysics?
The Tragedy of Set
Let us consider all objects which are not mountains, and imagine somehow that we can all put them in one basket. Will we need to put the set of all non-mountains in the basket also? How would we proceed? Let us take a goat, it is definitely a non-mountain. Hup, in the basket! How about despair? Certainly not a mountain. Go tickle the goat! 
The set of all non-mountains. Ho there! And who might you be? Set points at the tag on his jacket: The Set Of All Non-Mountains. Reluctantly you throw it into the basket, only to see it fly right back out and fall face first on the ground. The basket harrumphes and slowly quiets down.
You: What is it between you and the basket?
Set (patting himself vigorously all over in a cloud of dust): she thinks I am after her job.
You: aren't you?
Set: He! I can't help what I am!
You: what are you exactly?
Set: well, I am kind of a basket.
You: kind of?
Set: Not really. It's not like you can put anything in me, like in that stupid basket over there. Yes, You! I am more like a sign, or a pointing finger to other things.
You: so, you are not a non-mountain?
Set: I am too! Even you can see that, can't you?
You (annoyed): you do not look like a mountain to me, but they are known to be quite deceitful.
George: May I?
Both (sounding relieved): yes please!
George (to Set): can you point at other things besides non-mountains?
Set (indignant): Oh, I know what you're getting at!
George (shrewd): well, it is for your own good. Your reputation is not exactly stellar right now.
Set (takes a challenging pose, hands on the hips): okay, explain to me one thing. Can you have a set of sets?
You: who? Me?
Set (irritated): Not you, anybody!
George (very calm and sure of himself): sure you can. But why would you?
You: I don't know, I never tried it.
George (calmly walks you out of the room and closes the door behind you): You mean a finger pointing at other pointing fingers?
Set (belligerent): yeah,something like that!
George: what would the other fingers point at?
Set (unsure): nothing.
George: you mean that they point at nothing, that they are all empty sets?
Set: Yeah, they don't point at nothing at all.
George: There can be only one, Highlander. So, either you are the empty set, or you point at something.
Set: says who?
George: I'm afraid they all say that.
Set: how about class? I can have class too!
George: you would still be a set.
Set (in a mournful tone): they always get you in the end. So I guess I am not allowed in the basket, he?
George: what would you point at in the basket?
Set: I don't know, at everything in there? And also at myself? I am a non-mountain, whatever you and You say.
George: you would need your own basket, or at least, a place apart from all the others. Which would make you different from all the others. And that is why the basket didn't allow you in to start with. You're either in or out. See, you are not a non-mountain, or a non-anything. You are Set. You point at things, but you are not yourself a thing.
Set: so, when a mathematician speaks of different sets, and subsets?
George: it's always you, only you.
Set: I'm alone then.
George: I'm afraid you are, kid.

Set Theory: Mathematics or Metaphysics?
Some questions about: "how could a mathematical theory ever found mathematics? What would then found Set Theory itself? Its axioms? They are all of a mathematical nature, so that would not work. We would need non-mathematical axioms to found Set Theory, before it ever could found Mathematics."  1) The term "foundations" as used by logicians often means a set of axioms for a theory such that all the propositions of the theory (which include the axioms) are logical consequences of the theory. In this sense, the foundations for set theory must be mathematical.  Do you think this meaning of foundations is illegitimate? If so, why?
2) Suppose that someone says, "Here's a non-mathematical set of axioms for Set Theory".  That would be an interesting claim.  How would the someone prove his claim or even make it plausible? Would those non-mathematical axioms, if they are a foundation, reduce Set Theory to something non-mathematical?

Set Theory: Mathematics or Metaphysics?
How to build a set from scratch
It might sound like a strange question, but if you think about it for a minute you will see that it certainly deserves some close attention. After all, the only definition of set comes from Cantor, and it is quite elusive: it is an act of thought by which we consider different things as a whole. As the Tragedy of Set tried to illustrate, a set is not a basket. So what is it really?
Set theorists refuse to get caught in what Zermelo termed a philosophical discussion, the nature of sets. Instead, they let the axioms speak for themselves. Sets are what Set Theory says they are. That is also the position of "Second Philosopher" Penelope Maddy, the professional apologist of Science in general, and Set Theory in particular (see "Defending the Axioms", 2011).
I think it is a big mistake to let practitioners decide by their practice what a concept is supposed to mean. Mind you, I certainly do not advocate a Platonician government of philosophers who would then not only supervise, but also direct mathematicians in their work. But between Millikan Laissez-Faire and Plato's authoritarianism maybe a more balanced approach is possible.
First we must realize that Mathematics, not to mention mathematicians, is not inherently immune to metaphysical preconceptions. This should be obvious when we look at the way sets are treated in Set Theory. They function the way (abstract) baskets would. They have the same ontological status as their elements, and that is why the so-called paradoxes had such an impact. Even Cantor's pragmatic attitude relied on the same principle: sets are real, and the concept of "set of sets" is a problematic one.
What we can learn from Set is the following: Cantor's definition of sets is the (only) right one. They are a creation of the mind which, unlike numbers, which could also be considered as such, can never be separated from their creator. If we consider them as a finger used to point at other elements, or if you prefer, a flash light that illuminates specific elements each time, we avoid any semblance of paradox at a very low metaphysical cost.
Set Theory, especially post-Cantor variants, is best described as ad hoc policy in reaction to a crime already committed. Its axioms only make sense if we keep in mind, first, what they are trying to avoid, paradoxes, second, what they think they need to prove, the existence of specific sets.
When we consider sets as creations of the mind, we do not need to justify their existence. In fact, we could not. How do you justify ideas that come up in your mind? You can of course analyze them in a critical manner to find out if they are more than mere mental aberrations. But you certainly do not need to justify the fact that you can look at the world and center your attention on only one object (singleton), or two (pair), or a specific group with a property you deem interesting.
In a way Set is part of us [yes George, just like you], and ever present. Anytime we isolate an object or a group of objects from others, we create a set. That does not mean that our universe, whatever its ontological status might be, suddenly contains an extra new object. You might as well count the mathematician as also an object of that universe [and maybe you should!]. The only real aspects of sets are the (curly) brackets or parentheses we use to indicate which objects are now on our mind. They help us keep track of what we are doing with those objects.

The question now is, can Set Theory survive without sets?
First let me state that mathematician's freedom entails that mathematicians are free to conceive of sets as Real objects. My aim was only to show that it is neither evident nor necessary.
Second, even if my analysis is taken as correct, only mathematicians could answer the question. But it would certainly be interesting to observe such a new development.

How to build a set from scratch? You don't have to, just be yourself.

Set Theory: Mathematics or Metaphysics?
The idea that 0.999... is equal to 1 is a very old one, dating at least from Euler's "Elements of Algebra" of 1770. It goes something like this:
1) x =  0.999...
2) 10x= 9.999... 
3) 10x= 9+0.999...
4) 10x=9+x
5) 9x=9
6) x=1

Such a sequence or arguments only makes sense if we assume the reality of actual infinity, that is, if we could manipulate .999... like we would .999 (without ellipsis). It shows the power of deceit of the concept of actual infinity long before Cantor turned it into a cornerstone of his system.
When translated into a Limit framework we get the following result:

i) x=0.999... Lim x =1
ii) 10x=9.999... Lim 10x=10
iii) 10x= Lim 9+x
This is where it becomes tricky again. In fact, you cannot simply assert (iii) because 0.999... times 9 = 8.99...1, which is different from Lim 9.
iv) 9x = Lim 9 is false, even though both
v)x= Lim 1 and
vi) .999 = Lim 1 are true.

The only thing such an argumentation proves, even if I used the mathematical symbols the wrong way, is that it is not evident to apply arithmetical rules the same way to definite values and to open concepts like limits. I would consider it a strong caution against the unbridled use of the concept of actual infinity.
What I find most disquieting is the attempt to stigmatize students whose intuition make them reject the equation 0.999...=1. There are even psychological theories (see Ed Dubinsk and Michael Mc Donald, 2001, "APOS: A Constructivist Theory of Learning") which are supposed to explain why such students do not accept this equation. I find this approach, at the risk of being over-dramatic, a form of intellectual terrorism, worthy of the Gulag: you do not agree with us, then you must be sick!
There may be many legitimate reasons to accept the equation as a convention (see Timothy Gowers "Mathematics: A Very Short Introduction", 2002), but taking the validity of the argumentation at face value is unacceptable.

Set Theory: Mathematics or Metaphysics?
The Continuum revisited
Cantor's system relies on two proofs of infinity, the first is based on his diagonal method, proving that not all numbers sets are denumerable, the second on the so-called Continuum or density of the number line. It is usually presented as a triangle with a line parallel to the base and dissecting both sides. The proof makes appeal to our spatial intuition that every line crossing one parallel line will also cross the other, which makes the shorter line equivalent to the longer one, the base.
In the entry Reference and Geometry in The Liar Paradox (and other beasties),, I also used a geometrical argumentation to refute this idea. This will be no different, but I hope that it will strengthen my reasoning.
let us consider the triangle PCD with base CD and the parallel line AB cutting the sides PC and PD. 
What would it take for one line to cut AB at the same point and CD at different points? This is after all, the only way a smaller number of points could be put into a direct link to a greater number of points.
Unless of course we assume that there are as many points on the shorter line as there are on the longer one. But that is exactly what we needed to prove. 
We know that three points can be crossed by the same line only if they are collinear [at least in Euclidean geometry, and that is what is always used, as is the case in Fraenkel's "Abstract Theory", 1953, p.70]. But if each point of AB is collinear with a point on a line PC, then there will always be points on CD which are not collinear with any point of AB. Unless, once again, we assume what we are supposed to prove.
I will not try to turn this intuition into a geometrical proof, that is I'm afraid beyond my capacity.

If I am right, we do not need to appeal to a special spatial principle (one point one line) to arrive at the same conclusion. In this case, we are clearly dealing with a circular argument.
By magnifying the drawing sufficiently we would see that even when all points of AB are used, there are still points left empty on CD, although we do not see them normally because the distances between the lines are too small.
Furthermore, if Cantor's argument were correct Achilles would still be trying to catch up with the tortoise, which would also be still a long way from home after more than 2000 years!

Let us look now at what Fraenkel calls an analytical proof. Talking about functions, he give one specific example, y=tan x, and a general explanation: "By relating to each value of the abscissa x the value of the ordinate y, one obtains a one-to-one correspondence between the points of an interval and of the entire line." (p.73)
It certainly sounds more convincing than the geometrical proof, but is it also more valid?
It is obvious that to each x value will correspond a y value, even if if one part, the segment of x values, is allegedly shorter than the other, the line representing the tan function. I say allegedly because Fraenkel is in fact the victim of his own illusions. He thinks that he can equate all the values of x to the interval -pie/2 to pie/2, and in the same way, the values of y to an entire line. What he forgets is that the graph is only an approximation of the mathematical function. The values of y come only to existence when a value of x has been defined. Within the same space allotted by the graph both the values of x and those of y have to find a place. The more values we want to represent, the more fine tuned, or larger, the graph will need to be. The fact that we can squeeze more and more values in the same space is not a proof of infinity of a segment, only of the suppleness of our graphic representation, and of our own imagination. Let us not forget that graphs are only the concretization of abstract mathematical, thought processes. In short, Fraenkel's "analytical proof" does not really prove anything, besides the power of our own imagination. We can make a segment contain as many points as a whole line because at one point, we have stopped expanding the line and kept enlarging the segment. We can as easily reverse the process and make the segment disappear in nothingness while we expand the line beyond any imaginable border. Neither the first process nor its reverse prove anything about the properties of number lines and their segments, and all about what our mind is capable of. And what is the Continuum but a manifestation of that same power? I am personally not surprised at all that no formula concerning the Continuum could be proved: how, after all, do you measure an idea?

How many Transcendental Numbers are there?
According to Cantor, an infinite number. In fact many more than the algebraic numbers however counterintuitive that may seem. After all, we do not know that many transcendental numbers, and new ones are proved to exist only very rarely. So, how can we explain this discrepancy?
Still following Fraenkel, we learn that all depends on the following Theorem 1 (p.65; see also p.74 for the proof that the number of transcendentals is not denumerable because algebraic numbers are denumerable while the whole of the reals is not):
"The infinite set C of all real numbers between 0 and 1 is not denumerable."
That is what the diagonal method had proved: whatever real number between 0 and 1 we can try to put on a list, there will be always more where they come from.
A very powerful method that shows beyond any doubt the difference between the real numbers and other denumerable ones like rational, algebraic and natural numbers.
Still, I also pointed out the fact that each series of reals, at its own level, can be considered as denumerable. In this case Theorem 1 would not be much help to determine the number of transcendental numbers in each series. In fact, there would be no reason at all to believe that there will be more transcendental than, say, algebraic numbers at each level. Therefore, if we take the whole set of real numbers, level by level, it will be denumerable, and the apparent scarcity of transcendental numbers will be much more comprehensible.

How many Aleph's are there?
Still according to Fraenkel (1953), Theorem 1, "The infinite set C of all real numbers between ) and 1 is not denumerable", can be used to prove the existence of at least one more transfinite cardinal, different from aleph zero and aleph [he does not use here the index one as does Cantor]. Again, the diagonal method is called to the rescue: "the set F of all (real) functions f(x), x running over the interval from 0 to 1" (p.84ff) is defined, and the method shows that there always remain functions which have not been listed before, whatever we do. Since The continuum is equivalent to one of the subsets of F, this proves according to Fraenkel that the cardinal number of F is higher than C, and therefore different from both transfinite cardinals known until now.
Fraenkel's loyalty to Cantor is somehow touching even if I must admit being very often irritated at the incredible lack of logical sobriety from such a great logician and mathematician. His blindness to the weaknesses in Cantor's system remains a mystery to me. In this case, it regards the misuse and abuse of the diagonal method. Fraenkel, just like Cantor, considers that it proves the existence of a different cardinal than Aleph zero, and that is certainly far from evident. The fact that natural numbers and reals cannot be counted the same way can hardly be said to be the proof of two levels of infinity. Especially if we consider the fact that in both cases natural numbers are used to show the existence of a level which should be beyond their reach: Shaito san's magical bucket. This is apparently a matter of conviction that has nothing to do with Logic of Mathematics, and all with metaphysical conceptions of the Infinite. You either believe that there are different levels of Infinity or you don't. In the second case the diagonal method does not prove anything but the need to approach natural numbers and reals differently.
The mystical belief in the magical properties of the diagonal method is present again in the next step, where it is used to prove the existence of still another Aleph. But what holds for the first application holds certainly for the second. If not more. After all, the same number sets are used to justify once again one more level of infinity. In principle Fraenkel could go on indefinitely: take the level he has allegedly reached, and show by the diagonal method that the next level can be proven to exist exactly the same way as the ones preceding it. Ad nauseam.
Any arbitrary definition of infinity could be shown to be insufficient for the next level: that is the power of the diagonal method. To use it as the proof of different levels of infinity is, in my eyes, to abuse it.

Set Theory: Mathematics or Metaphysics?
Cantor-Bernstein Theorem viewed by Fraenkel
Fraenkel gives a proof of this (in)famous theorem that is quite interesting. It is quite a long proof, so the reader will pardon me if I do not quote it in its entirety (see p.99ff). I will try though to unveil its structure as well as I am capable of.

[~ means equivalent]
Problem: If a set S is equivalent to a proper subset T1 of a set T, and T is equivalent to a proper subset S1 of S, then S and T are said to be equivalent.
Notice that in this version Fraenkel is speaking exclusively of proper subsets, which makes my linguistic approach not applicable here. At least, not at first. (see the entry "Cantor-Bernstein Theorem: a real or apparent problem?" in The Liar Paradox (and other beasties))

S~T1 proper subset of T
T~S1 proper subset of S
We want to prove: S~T.

Fraenkel gives a proof in two parts, so we will follow each of his steps very closely.

First part of the proof
Fraenkel uses the known Cantorian strategy of creating an intermediary level that takes all the punches, but also guarantees victory: he takes the relation T~S1, and creates what is supposed to be also a valid relation between the same [this is essential!] T1 that was used in the relation with S, and a subset of S2 of S1.
We now have an extra relation we can use:
T1~S2 with S2 being a subset of S1
It seems quite innocuous until we realize that such a relation gives rise to:
S1~T1, T1~S2 => S~S2.
Which is quite strange, since S2 was first introduced as a subset of S.

Let us recapitulate before we get lost in this jungle of sets and subsets:
a) S~T1  b) T~S1
c) S~S2   d) T1~S2
The first two, (a) and (b) were given, Fraenkel has added (d), and that gives us (c) for free. What should be considered as a contradiction is here deemed acceptable, since infinite sets are allowed to be equivalent to one, or more, of their subsets. It also shows why it had to be T1 again that was used and not any arbitrary subset T2. Fraenkel follows his Guru in assuming the problem solved at the intermediary level, which makes it trivial to prove its general validity. An arbitrary subset would not have brought anything new to the situation and would have therefore only made it unnecessarily complicated.
Assuming the argumentation valid, which it is certainly not, what has Fraenkel accomplished so far?

Second part of the proof
We still need to prove S~T, and for that we need S1~S2, or S~S1, to get to the final prize, S~T. For that, Fraenkel expresses the result of the first part of the proof in a new theorem which I will sum up as: if we know that a set S is equivalent to its proper subset S2, then we can show that it is equivalent to all its other subsets S1 which have S2 as their proper subset.

I am afraid I cannot present you with a drawing, so I will describe the situation to you.
Take a big circle, and call it S. Within that circle draw a smaller circle S1, then within S1 a final circle S2.
You will understand the difficulty Fraenkel and all the others have been facing with this theorem when you picture a, more or less, equivalent set of circles for T. We have now to prove that the fact that one of the big circles is equivalent to one of the smaller circles of the other party, makes both big circles equivalent. My own humble opinion is that such a proof is simply impossible, that it concerns here an empirical and not a logical fact. Let us see how Fraenkel handles himself in such a perilous situation.
As stated before, all Fraenkel needs to do now is prove that S is equivalent to all the proper subsets which include S2 as a proper subset. In other words, if the big circle is equivalent to the smallest circle, it is also equivalent to the middle circle.
For that Fraenkel switches gears and, instead of picturing circles you have now to imagine the different parts as segments, and this is what it is supposed to look like [I have added D to make the adaptation of the drawing more compatible with the original. See p.101]:
A                               B             C
A1                   B1     C1
A2          B2  C2
A3    B3 C3

As you have probably already guessed, A (segment AD) represents the big circle S, B (BC) is S1, and S2 is C (CD).
And now the big magical trick:
Fraenkel reformulates the theorem this way: we have to prove that 
Fraenkel "shows" that A is equivalent to all its subsets A1...Ak, just like B and C are equivalent to respectively B1...Bk and C1...Ck
Also, we can reformulate B+C as the preceding A+B+C. Which makes it possible for us to put A+B+C in a one to one correspondence with A+B. Which is also a very nice trick, albeit not entirely kosher. And that is all we need to prove our theorem.
How does Fraenkel show such a one to one correspondence between the different subsets? Not difficult at all, it is true by definition. Each set is equal to its proper subset, ad infinitum.
We see that the whole proof hinges on the validity of this Cantorian dogma, and that brings me to my final remark.
The emphasis on proper subsets ends up having very little substance, it has simply been replaced by the principle that a set is equivalent to its subset. Which of course comes down to the same: the sets are in fact declared to be equivalent to each other by definition, be it indirectly via their subsets. 
Fraenkel has certainly not proved that the Cantor-Bernstein Theorem is not a pseudo-theorem.

Set Theory: Mathematics or Metaphysics?
Fraenkel Goes to Hallewood
"Theorem 4. A finite set is not equivalent to any proper subset of itself."
Let me first state my bewilderment at the idea that such a principle is in need of a theorem. After all, it comes down to the, obvious, Euclidean principle of the whole being greater than its parts. Something which was taken as an axiom by the Greek geometrician, and therefore unprovable. Of course, within the Cantorian system it makes perfect sense: if the Euclidean principle is untrue when it comes to infinite sets, then we are allowed to see the equivalent principle concerning finite sets as something that needs proof also. In other words, this theorem only makes sense in a Cantorian perspective. That means we are already assuming that infinite sets can be equivalent to their subsets.
Let's try not to let this point out of our sight.

First allow me to sum up the proof, by mostly paraphrasing it:
If S is a set equivalent to the set of integers  {1, 2, . . ., n, n + 1} elements [...] we may denote the elements of S by s1 s2, ...,sn, sn+1 [...]. We suppose the existence of a proper subset S' of S that is equivalent to S, and infer from this that also the subset {s1s2, . . ., sn} of S is equivalent to a proper subset of itself — which contradicts our assumption.

In other words, Fraenkel, (just like Cantor likes to do) introduces an intermediary step in the form of a subset of the subset!

0) "The theorem is true if the set contains only one element; for  then the only proper subset is the null-set which, containing no element, cannot be equivalent to a set containing one element. "
This is the basis of the proof. Its evidence, 0#1 does not need anything else to make it acceptable. From here it sounds reasonable to extend the number of elements of S to an arbitrary number n. But does the theorem remain valid when we get to n+1? 
"Let us assume the theorem to be true for all sets of n elements, n being a certain natural number whatsoever."
Fraenkel sees three possibilities, all concerning the status of the extra element sn+1. We will analyze them one by one as does the author.

First Possibility
"a) The subset S' does not contain the element sn+1 of S. Then the mate x  of sn+1 ε S in S' is different from sn+1. Let us write S' = S" + {x} where S"  is a proper subset of S' . Therefore, when we remove sn+1 ε S and its mate x ε S' from the supposed representation between S and S', there remains a representation between {s1, s2, . . ., sn} and its proper subset S", contrary to our assumption that for a set of n elements no such representation exists. 

i) "Then the mate x of sn+1 E S in S' is different from sn+1."
gives us:

Notice how biased Fraenkel's interpretation is. S' does not differ from S by the number of its elements, but by their nature! The presence of x is a mystery. What is it, and where does it come from? As true of any x that respects itself, it remains an unknown.

ii) "Let us write S' = S" + {x} where S" is a proper subset of S'."

So, S" does not rate a mystery element y. It is plain S'-x.
To be consequent, Fraenkel should have said
with y#x and therefore y#sn+1.
The problem of course would have been that each subsequent subset would have had to contain its own mystery element, making subsets all as large as their containing sets. Which makes the following statement false!

iii) "Therefore, when we remove sn+1 ε S and its mate x ε S' from the supposed representation between S and S', there remains a representation between {s1, s2, . . ., sn} and its proper subset S", contrary to our assumption that for a set of n elements no such representation exists."
Normally, following his own logic, Fraenkel would have to remove also y from S", z from S"'..., and so on ad infinitum!
The arbitrary use of an intermediary level with its mystery element makes the conclusion stand on shaking grounds. Fraenkel did not prove the obvious!

Second Possibility
"b) S' contains sn+1, and sn+1 ε S is attached to sn+1 ε S', i.e. to itself. After dropping the element sn+1 out of the supposed representation between S and S' in both sets, we again get a representation between {s1; s2, . . ., sn} and a proper subset of this set, contrary to our supposition."

We get the following:
S ={s1,s2,...,sn,sn+1}

What does the proof amount to? If we drop sn+1 from both S and S', we will get a representation, or one to one correspondence between S and S', contrary to our assumption that they are not equivalent.
How could Fraenkel print such an absurdity? We do not need to drop sn+1 to see that in this case S and S' are equivalent, contrary to the assumption! This second possibility is a joke pure and simple!

Third Possibility
"c) S' contains sn+1; but in the presumed representation between S and S', sn+1 ε S is attached to another element of S', say to y, while the mate of sn+1  ε S' in S is a certain element x ε S (x = y not exluded) [sic]. We thus have the scheme of correspondence: sn+1 <-->y, x <—-> sn+1."

S ={s1,s2,...,sn,sn+1,x}

In this scenario, x has moved from S' to S, but not before begetting y and leaving it behind immediately. We then get something like cross-pollination with sn+1  in S copulating with y and sn+1in S' preferring x on the other side. Again, x and y cannot be considered as (natural) numbers, and we are therefore left in some kind of limbo as what their nature does to the relation between S and S'. It would be like saying:
S ={s1,s2,...,sn,sn+1,apple}

The following step is a stubborn refusal of such an unnatural situation:" Now, we modify the representation between S and S', by relating sn+1 ε S to sn+1 ε S' and x ε S to y ε S'. Thus, we get a new representation between S and S' in which sn+1 ε S is attached to sn+1 ε S'. But this is the case b), already found contradictory."
In other words, we do like x and y never existed. But then why introduce them in the first place?

Fraenkel here proves himself a worthy disciple of Cantor. His logic in this proof is Cantor's logic at its worst. He tries to prove a statement that had been considered as obvious millenia before Cantor's revolution. His failure to do so is is mirrored by Cantor's failure to prove the existence of the Aleph's on pure rational and logical grounds. 

Set Theory: Mathematics or Metaphysics?
The Surprising Adventures of Baron Münchhausen may somehow give an answer to your questions. He pulled himself and his horse up out of the swamp by his own hair. That is in fact what any mathematical theory would do trying to found mathematics.
I would not say the question of the foundations of mathematics is an illegitimate question, only that it is not a mathematical matter. We are entering the (Aristotelian) realm of "first causes" and they certainly do not belong to Mathematics, however strongly some authors wish they were.
That does not mean that set theory cannot help reformulate other mathematical theories in another language, maybe more amenable to other types of proofs. But it would be a reformulation, not a foundation. 
So yes, maybe set theory can replace all other mathematical theories, but it would still not explain how mathematics are possible. Which brings me to your second question.
There would be nothing strange to someone presenting a non mathematical theory of set theory or any other mathematical theory. In fact, the literature is full of attempts to do just that, starting with Plato and Aristotle.

Set Theory: Mathematics or Metaphysics?
En Attendant Gödel
Gödel's Approach remains the same both in his Incompleteness and his Consistency Proofs
1) take the statements you are interested in,
2) build a logical model of these statements, 
and then
3) prove one of two possibilities, according to your objective:
a) there are statements in the model that do not appear in the original set,
b) all statements of the model are  acceptable according to whatever criteria you have, and are therefore valid.

The problem of course plays on two fronts: the model itself, and the versatility of the method.
Concerning the last one, the diagonal method can be used to prove that any system comes short of its own objective, even to the point of contradicting itself.
Second, why do we need a model to prove the validity of the statements or theory under analysis? Why not deal with them directly? After all, it really does not prove anything that could not be proved by the analysis of the theory itself. Not if it is a good model anyway.
It seems that Gödel's approach is very opportunistic. He uses the diagonal method when he wants to discard a theory, but avoids it completely when his aim is to support the theory under scrutiny.
I find it philosophically very suspicious that Gödel each time seems capable of mechanizing the process of drawing value judgments. If that were possible, then it would be reasonable to let machines decide of not only what is True, but also of what is Beautiful and Good. When we realize that Gödel was a convinced Platonist, we understand that such a possibility did not frighten him. He did not need God like Cantor did. A (logical) machine could do the job as well.

Set Theory: Mathematics or Metaphysics?
On Wikipedia, the article "0.999..." presents a great many proofs that 0.999... is equal to 1.  I suggest that the following, taken from that article, is conclusive proof:

Q: How many mathematicians does it take to screw in a light bulb?

A: 0.999999….

Set Theory: Mathematics or Metaphysics?
You misunderstood my second question, which was: 
Suppose that someone says, "Here's a non-mathematical set of axioms for Set Theory".  That would be an interesting claim.  How would the someone prove his claim or even make it plausible? Would those non-mathematical axioms, if they are a foundation, reduce Set Theory to something non-mathematical?

Let's suppose that you extract views about sets from the writings of Plato and label those views non-mathematical axioms for set theory.  How would you prove that they are plausible axioms for set theory?  Would you, for example, formalize them in a suitable formal language and, using suitable axioms, deduce ZF from them? 

Set Theory: Mathematics or Metaphysics?
"Here's a non-mathematical set of axioms for set theory". I am not sure that I understand it at all. A philosophical foundation of axiom theory, if it is even possible, would not present axioms for set theory, but for me to say what it would do would suppose that I already know how such a foundation would look like. I will humbly admit my ignorance in this matter. It is much easier to say: foundation of mathematics cannot itself be mathematical, in a general, and safely abstract way, than pinpoint the concrete properties of such a foundation.
If I understand you right, then I agree that non-mathematical axioms could never replace the mathematical axioms of set theory. A philosophical foundation which would get down to such a detailed level would probably simply end up replacing one theory with the other, and loose any claim to being a foundation.
All this, under the assumption that it is even possible to build such a foundation. I prefer to look at it as a work eternally in progress, where we learn as we go and never stop learning.
I still believe though that the foundational pretensions of set theory are unjustified. Set theory is just a theory like any other mathematical theory, whatever its advantages might be over the others.
As I have tried to show, it certainly can no more explain "number" than one to one correspondence or numerals can, since they all can be considered as different forms of expressions of "number". And such an explanation would be indispensable for any theory that would pretend to the title of "foundation".
But whether a non-mathematical theory would itself be capable of defining "number"? That is the big question.
All in all, what is said of set theory can generally be relegated to the metaphysical realm where it belongs. Many mathematicians are in fact bigots who believe in the infallibility of the mathematical Pope, even if individual Cardinals may stray. If we consider mathematics as a human language, and not as a divine tool (as Galileo and all those following him did) , then we must accept that, even with all its advantages over natural languages, it remains nothing more but a language, and will therefore only express what mathematicians, themselves humans, can think about different subjects. I think I have shown often enough that formulas and symbols do not guarantee that the mathematician will be free of the same metaphysical prejudices that philosophers are known to suffer under. The advantage of philosophers is that their own prejudices form very often themselves the subject of their research, while mathematicians can hide behind their symbols and never face their own metaphysical pre-conceptions.
You could of course mention Platonists, Intuitionists and Formalists, speaking only of the major schools, but I think that it goes beyond all those party lines. These differences are built on a consensus: the superiority of the mathematical over the natural language. I think that this is the most unjustified and dangerous delusion of all. The advantage of the natural language is its openness, even if it is regularly abused by authors whose style is so obscure that they might as well use a symbolic language. Mathematical language is perfectly adapted to its object, mathematics, but it does not in itself guarantee truth.
The mystique surrounding the language is such that whole branches of knowledge are corrupted by the myth that to express something in a mathematical formula is to make it scientific. This is, I repeat, a very dangerous prejudice.

Set Theory: Mathematics or Metaphysics?
Hrbacek and Jech Also Go To Hallewood
This is another version of the "proof" that a finite set cannot be equivalent to its subset. Here is how the authors formulate it (p.70):
"Lemma If n element of N, then there is no one-to-one mapping of n onto a proper subset X C n. 
Proof. By induction on n. For n = 0, the assertion is trivially true. 
Assuming that it is true for n, let us prove it for n + 1. If the assertion is false for n+1, then there is a one-to-one mapping f of n+1 onto some X C n+1." 
This is very similar to the first part of the proof by Fraenkel, and I will therefore skip it.

They continue: 
"There are two possible cases: Either n element of X or n not-element of X.
If n not-element of X, then X subset of n and f|n  maps n onto a proper subset X - {f(n)} of n, a contradiction." 

I am not sure I understand it right. This is what I see:
The authors are trying to prove a reductio, that n is equivalent to its subset X. They take first the case that n is not an element of X, in other words, n is greater than X. At least, that is how I would interpret "n not-element of X" if we are dealing with sets of (natural) numbers. But then the following statement seems to belie this interpretation. 

let us say n=X+1, the way I would interpret n not element of X.
Then X-1 is certainly a proper subset of n. But how did we get there?
X - {f(n)} certainly leads to a contradiction if we try to map it with n [into, onto, or whatever position you prefer], but I have no idea where this subset comes from!

Let us bravely continue our trek.

"If n € X, then n — f(k) for some k < n. We consider the function g on n defined as follows: (imagine both curly brackets as a big one) 
g(i)= {f(i) for all i # k, i < n; 
        {f(n) if i = k < n. 
The function g is one-to-one and maps n onto X — {n}} a proper subset of n. a contradiction." 

I will assume that the authors are right, and that this leads to a contradiction. But then, so what? The contradiction it leads to is that a set is put on a one to one correspondence with its subset. But if it is a contradiction, why start the proof at all? They wanted to prove that "there is no one to one mapping between a set and its subset", and all they showed for it was that, whatever they did, they arrived at the wrong conclusion. In fact, the only correct conclusion on the basis of their argumentation would be that, against all odds, there is a one to one mapping between a finite set and its subset!

Just like Fraenkel, and Cantor before him, Hrbacek and Jech want to prove the obvious, a millennial axiom, and simply fail to do so.

Set Theory: Mathematics or Metaphysics?
Cantor-Bernstein Theorem viewed by Hrbacek and Jech
("Introduction to Set Theory", 1999; p.66ff) 

"If |X| ≤ |Y| and |Y| ≤ |X|, then |X| = |Y|."

Let us first remark that this formulation of the theorem makes it possible for me to apply the linguistic approach and immediately declare the problem as a pseudo-problem.  (see the entry "Cantor-Bernstein Theorem: a real or apparent problem?" in The Liar Paradox (and other beasties)
Still, let us see what the authors have to offer with their magical formulas.

 If |X| -< |Y|, then there is a one-to-one function f that maps X into Y; if |V| -< |X|, then there is a one-to-one function g that maps Y into X. To show that |X| = |Y| we have to exhibit a one-to-one function which maps X onto Y."

Given: f and g are injections (into, or one to one function). At most one y is linked to an x, and vice versa. There may be free x's and y's floating around.

We want: (I have made the link longer to indicate that it is two ways)

"Let us apply first f and then g; the function g o f maps X into X and is one-to-one. Clearly, g[f[X]] subset of  g[Y] subset of X"
This seems like a smart move, except for the following objection:
The composition function g o f gives us, taking a concrete example, y1_x1, which allows us to deduce x1__y1. But that is exactly what we wanted to prove! To assume it is true because it is mathematically possible is unacceptable. And that is without taking into consideration the free x's and y's.
This first step is therefore an illegal move!

He then makes the theorem follow from this lemma:
"If A subset of B subset of A and A1 = |A|, then |B| = |A|."
The strategy is the same as by Fraenkel: the set A is declared equivalent to its subset A1.
But what is really telling is the diagram which is supposed to make the proof intuitively palatable. It is the drawing of a big circle (A), with an inscribed square (B), itself containing another circle (A1), with square B1, and so on. What has started as the relation between two distinct sets  X and Y, has degenerated into the relation between a single figure and its subsets, the difference between circles and squares being purely cosmetic!
The conclusion is therefore inevitable, A=B. But how could it be any different when everything was reinterpreted in such a way as to make that possible? The same trick is used as by Fraenkel: first assume that a set is equivalent to its subsets, and then turn the other set to one of the subsets, and voilà!
I wish I could reproduce here the beautiful formula that the authors arrive at. You can admire it at p.67. 
It might as well have been a pentagram to summon Shaito san. I would not count on him being in a good mood though.
[I wonder if Cantor and Bernstein never published their own proofs because they were smart enough to see that it simply was not possible to give one? The reason usually given is that Bernstein's own proof was probably not as good as Borel's. But how come Cantor never tried his hand at it? Maybe they were both happy enough to take the credit without incurring the risks of being proven wrong.]

Set Theory: Mathematics or Metaphysics?
Very useful perspective for a philophosical question of mine! I am against the thesis of aktual infinity similar to the answers of Leibniz or Kant, and sometimes I look a little bit to Cantor, Brouwer, Hilbert, Zermelo. But the most of the time I recognize only weak analogies to Ideas about segmentation of semantics in Logic after the "linguistic turn" without genuin understanding in mathematics.

Thanks, Wolfgang

Set Theory: Mathematics or Metaphysics?
I can agree in many points. But perhaps You try to know something about the difference between »unvollständige Summendefinition« and »vollständige Summendefinition« from Bolzano, Wissenschaftslehre II, which deduced the natural numbers with the logical präsuppositions of using language. Its come from Kant and Baumgarten (Mereology). — At least, You also speaking about language. — I think, unable to demonstrate it formally as logic or mathematics, that some part of the präsuppostions of mathematics are linguistic and logic, and that some parts of the präsuppositions of logic are mathematics and linguistic. Set theory (in my simple way) could be the first step for an analytic proof of contradictions, but is not able to found mathematics or logic at all.

Set Theory: Mathematics or Metaphysics?
Cantor-Bernstein Theorem (CBT): much ado about nothing?
[I thank Eray Ozkural for this movie reference. I did not know the movie, (and I still do not), so I did not completely catch his sarcasm the first time.]

My analysis of this theorem is apparently far from original. In fact, it had already been presented 110 years ago by Hessenberg in his "Grundzüge der Mengentheorie", 1906). What makes this author very interesting, besides his other mathematical achievements, is that, two years before Zermelo's second proof and first axiomatization of set theory he tried to build such a system. 
I liked his warning about mysticism in Mathematics concerning Set Theory (see the introduction in "Das Unendliche in Mathematik", 1904), I just regret that he was not more critical of the whole project.
I will certainly come back to Hessenberg and his presentation of set Theory, let me just sum up, without going into any details, his view of CBT:
1) If one accepts the principle of the whole greater than the parts, then CBT is invalid.
2) When one takes said principle as not applicable to all cases, especially infinite sets, then CBT becomes valid.
3) A and B are equivalent when A equivalent to B1 subset of B, and B equivalent to subset A1 of A.
4) Principle (3) can be replaced by: 
If A1 is a part of A, A2 a part of A1, and A2~A, then A1~A.

He considers (4) as a better formulation of CBT than (3), since it involves only one set and its subsets.
What we can learn from this is that CBT can never be used as an argument in favor of Infinity since it can only be considered valid if we accept beforehand the definition of an infinite set as one that is equal to its subset. It cannot therefore be used as an argument in favor of any Cantorian conception since it is itself a consequence of Cantorian assumptions concerning the infinite.

Also, it is interesting to see that all the authors that I have analyzed so far make it sound like CBT is a product of pure logic, instead of being the result of a metaphysical assumption!

Set Theory: Mathematics or Metaphysics?
The Pentaquarks Debate is something way over my head. I tried to read "Observation of J= p resonances consistent with pentaquark states in ... decays" (2015) but I could not even understand the title! That does not mean that I did not learn anything from it. Allow me to express myself in general terms and try not make a fool of myself:
1) There is a set of physical data P with properties/elements p1,p2,..;
2) There is a set of mathematical data M with elements/properties m1,m2...;
3) Mathematical tools are used to analyze M and draw conclusions concerning the physical set P.

[I will leave the question aside whether both sets can even be distinguished from each other.]

The question now is: how can elementary particles scientists profit from Zermelo's insights concerning the well-ordering of an arbitrary set? Would their work be more difficult if set theory had never been invented?
I will take an uninformed stand and declare that it is highly implausible that CERN scientists and other teams would care one way or another about the validity of the Axiom of Choice. Their problem is, I think, that the data is so complex that finding an ordering principle for all elements would be like finding the answer to the "Question of Life, the Universe, and Everything" [and we already know it is 42, so that would certainly be a waste of time].
The scientists have therefore to settle for subsets of all the data, and how they come to these partitions will probably go way beyond some simple mathematical choices, and all have to do with knowledge and experience with how accelerators function, and how data is gathered.
We can of course assume that the Ultimate Answer will be somewhere hidden in the data provided by the accelerator. Pentaquarks for instance were (thought to be) found in data sets of a "retired" accelerator. And who knows what future scientists will be able to deduce from the same data sets scientists have at their disposition today.
But if that is what Zermelo's axiom of choice means, then its practical and theoretical usefulness is simply null. The problem is that I do not see how else it could be interpreted.

Set Theory: Mathematics or Metaphysics?
Cantor-Bernstein Theorem (CBT) viewed by Hessenberg

My analysis of this theorem is apparently far from original. In fact, it had already been presented 110 years ago by Hessenberg in his "Grundzüge der Mengentheorie", 1906). What makes this author very interesting, besides his other mathematical achievements, is that, two years before Zermelo's second proof and first axiomatization of set theory he tried to build such a system. 
I liked his warning about mysticism in Mathematics concerning Set Theory (see the introduction in "Das Unendliche in Mathematik", 1904), I just regret that he was not more critical of the whole project.
I will certainly come back to Hessenberg and his presentation of set Theory, let me just sum up, without going into any details, his view of CBT:
1) If one accepts the principle of the whole greater than the parts, then CBT is invalid.
2) When one takes said principle as not applicable to all cases, especially infinite sets, then CBT becomes valid.
3) A and B are equivalent when A equivalent to B1 subset of B, and B equivalent to subset A1 of A.
4) Principle (3) can be replaced by: 
If A1 is a part of A, A2 a part of A1, and A2~A, then A1~A.

He considers (4) as a better formulation of CBT than (3), since it involves only one set and its subsets.
What we can learn from this is that CBT can never be used as an argument in favor of Infinity since it can only be considered valid if we accept beforehand the definition of an infinite set as one that is equal to its subset. It cannot therefore be used as an argument in favor of any Cantorian conception since it is itself a consequence of Cantorian assumptions concerning the infinite.

Also, it is interesting to see that all the authors that I have analyzed so far make it sound like CBT is a product of pure logic, instead of being the result of a metaphysical assumption!

Set Theory: Mathematics or Metaphysics?
Simply Ordered (2)
We read in "Set Theory and Related Topics", 1998 by Seymour Lipschutz: "Consider the set Q of rational numbers with the usual order. Even though Q is linearly ordered, no element in Q has an immediate predecessor or an immediate successor. For if a, b E Q, say a < b, then (a + b)/2 belongs to Q and 
a < (a+b)/2 < b." (p.206)

When you consider the set of even numbers, you could certainly say that 4 is the immediate successor of 2. You do not worry about the fact that between 2 and 4 exists an infinity of reals, and one whole number, 3. Why should you then worry about what could possibly lie between 1/3 and 1/4 according to different definitions? The only criterion that "counts", is the one you want to apply to the situation. That means that you can really say that every number has an immediate successor or predecessor, be it natural, rational or even complex or imaginary. It just depends on what you want. In fact, such a principle would get along just fine with Zermelo's principle of well-ordering of any arbitrary set. After all, we are the one who define what "order" is supposed to mean in every situation.
It might seem like a philosophical position that should be avoided by mathematicians, and the first part is undeniably right, the question being whether mathematicians do always avoid this kind of decisions, or only do it when it seems convenient to them.
Let us look at a so-called proof which immediately followed the preceding quote, something students learn in school or in college.

"Theorem 8.5: Every element in a well-ordered set A has a unique immediate successor except the last element.
Let a E A, and let M(a) denote the set of elements of A which strictly succeeds a. If a is not the last element, then M(a) # O. Since A is well-ordered, M(a) has a first element, say b. We claim b is an immediate successor of a. Otherwise, there is an element c E A such that a < c < b. Then c E M(a) and this contradicts the fact that b is the first element of M(a). We claim b is the only immediate successor of d. Otherwise, there is another immediate successor of a, say d. Then d E M(a) and, since b is a first element of M(a), we have a < b < d. This contradicts the assumption that d is an immediate successor of a. Thus the first element b of M(a) is the unique immediate successor of a." (p.206)

When reading this proof we soon realize the central role played by the principle that a well-ordered set has a first element. Since it is a matter of definition, we have no right to object to the argument that whatever contradicts the definition cannot claim the title of well-ordered set. Still, let us see if the proof is valid anyway.
The rule is simple: if there is an element c before any b, then b cannot possibly be the first element, and since that was our initial assumption then c simply cannot exist. Unless of course we decide that our definition of first element is untenable. This definition assumes that we must be always able to determine when and whether an element can be considered as first element. At the same time, we do not find it strange that we are convinced that we cannot point at an immediate successor or predecessor of a rational number. What makes us use one different measure for each situation? How come we find it normal to be able to go from 1 to 2, and back, but not from 1/3 to 2/3? What stops us in the second case? And why doesn't it stop us in the first one?
If you asked a layman what comes after 1/3, you would probably get 2/3 as an answer (or 1/4). Why would it be a wrong answer?
This is I'm afraid the same point I have tried to made in the entry "Simply Ordered?": you cannot claim, on one hand, that numbers have no predecessors or successors, and on the other hand, act like the rule does not apply "normally". For that, you would need to explain how we can pass from a "normal" situation, where we can simply go from one number to the other, to a situation resembling Achilles' nightmare that every step he takes seems to broaden the distance between him and the Tortoise, or at at least, not bring the damn beast any closer.
After all, nothing compels me to say 2 after one, or stops me from saying 1,1 after 1, followed by 1,01, etc. 
So what it is it that makes one step legitimate in the eyes of mathematicians, but not the other?
I can therefore only conclude that the textbook proof is built on philosophical, not to say metaphysical assumptions that are completely arbitrary. We would be just as well justified to say that this proof shows that whatever we do, we can never reach the first element, just like Achilles can never reach the tortoise.

Set Theory: Mathematics or Metaphysics?
in von Neumann's "Zur Einfuhrung der Transfiniten Zahlen", 1923, we encounter a very interesting argumentation form, which I have the impression is used by many authors. Allow me the following free translation (see III.Kapitel section 10, par.2):
"Let X be a well-ordered set. Two counts f(x) and g(x) of X are identical.
Otherwise, there would be a first x for which f(x) # g(x). For all y -< x we would have f(y) = g(y), and therefore
M(f(y); y E X, y -< x) = M(g(y); y E X, y -< x) [y belongs to X, y precedes x]
which would give us f(x) = g(x), contrary to the assumption."

I find the argument: if f(x) is not identical to g(x), then "there would be a first x for which f(x) # g(x)", absolutely amazing.
What does it really mean, if it means anything at all?

- If two lines are not identical, then there must be a first point for which both lines are different.
- If two sets are not identical, then there must be a first element for which both sets are different.
- If two functions are not identical, then there must be a first element for which both functions are different.

In other words, the difference can only be temporary or partial. Once you have established when or where the difference starts, you have at the same time established when and where the identity of both categories start. And once you have done that, then the assumption of different categories, in our case functions, becomes suddenly a contradiction.
It is almost, if not certainly, Hegelian in its dialectic approach: we want to prove identity, and we do that by showing that difference itself gives rise to identity!
But is it wrong of von Neumann to think like that? let us see.
He wants to prove the fact that however we count the elements of a well-ordered set X, we will get the same ordinal number which he expresses in the identity f(x) = g(x).
I have already pointed out the impossibility of proving the identity of two, or more elements. The only way to know that a and b are identical is simply to see it, or believe it because of whatever reason. For instance, we know a = c, and we also know that b = c, so we have good reasons to believe that a = b. That was one fundamental mistake in Gödel's Incompleteness proofs, the idea that "evident" truths" could have metalogical equivalents which could then be translated into Gödel numbers. (see the entries "Gödel's proof in "On formally Undecidable Propositions of Principia Mathematica and Related Systems"" and "What is Diagonalization good for?" in The Liar Paradox (and other beasties)).
If my analysis is correct, then von Neumann cannot possibly prove what he wants, and there must be a mistake somewhere in his argumentation. Let us look at it again even more closely.
"there would be a first x for which f(x) # g(x)" is obviously the pivotal point in the whole argumentation. It also assumes that we are somehow capable of distinguishing the moment where identity goes over into difference, and vice-versa. Which I am sure we are, except that this also shows at the same time that we already must have a sense of the identity or difference of the two functions, otherwise, we would never be able to point at the locus of transition. By trying to prove the identity of the two functions, von Neumann is therefore already relying on a sense of their identity or difference. 
It certainly does not look like at it at a superficial glance, by his argumentation is definitely circular, and therefore vitiated.

Set Theory: Mathematics or Metaphysics?
Thank you for your suggestions. Also, I would like to point out that you are welcome to react in German. Just add a Google-translation if you can.

Set Theory: Mathematics or Metaphysics?
Cantor and Brouwer, Infinity and Black Holes
What do they have in common? Let us start with the last one. It is nigh a dogma of modern cosmology that the laws of time and space cease to function within a black hole. The problem of course is that we have no way (yet?), of knowing it one way or another, which makes it a matter of pure belief. Take the notion of Time for instance. It is supposed to slow down to the point that what would take a few seconds outside of a black hole, would last for years within. Which makes sense if we consider the fact that atoms and particles are slowed down considerably by increasing gravity. It would be like listening to the same recording at normal speed, while playing it on another device at a much lower speed. Imagine yourself a part of the imaginary world represented by the second device. You are speaking, thinking and acting, all in slow-mention, as far as an external observer is concerned, but you do not know it. Everything seems to happen like it always does outside of the black hole. This of course assuming that your can survive under such conditions, and that your brain can still function at such a slow speed.
And that is I am afraid, the big question.
Suppose that it does not. Suppose that long before entering a black hole, a living creature would simply be torn apart in atoms and electrons. Well, we could fall back on the Einsteinian picture of synchronized watches outside of and within the black hole. Would that change anything to our ignorance? We would still have to assume that the watch in the black hole does survive in there. And if it does, that would mean that it is not subject to the same laws that surround it. In other words, it would be useless to us as a source of information.
A favorite theme of sci-fi is cryo-sleep. Take a nap and wake up years or even centuries later. An effect very similar to what we are supposed to experience after a short dive in a black hole. Meanwhile, while we were asleep, the world kept to its boring routine of sunrises and sunsets.
Still, suppose we could somehow not only survive such a dive, but also remain conscious? Well, in this case, I am afraid that the results would be very disappointing: we would have to build a small copy of the physical laws as we know them in which we could survive and observe what is happening outside of them. A black hole submarine as it were. Why disappointing? Because the watches aboard would go as fast as the watches outside of the black hole.
But what about the experiment with watches aboard a fast flying plane and watches on the ground? I see no reason to doubt of its validity. After all, we are speaking of the effects of speed on matter, and therefore not only on the plane itself, but also on everything it contained. It would be like immersing a watch outside of the black hole craft and bring it back (against all odds) in one piece. No doubt it would show a different time than other watches. Once we leave the black hole, we would probably find out that time has passed as fast or as slowly for us than for all other people. The only difference would then be the fact that we had been witness to beautiful scenes of particles moving very slowly around.
What about Cantor and Brouwer ("On the significance of the principle of excluded middle in mathematics, especially in function theory", 1923)? Both were convinced, just like modern cosmologists, that Infinity has its own peculiar laws. Brouwer never dreamed of denying the validity of the law of the excluded middle for finite sets, only for the infinite. The same way Cantor was convinced that some perennial laws, like that of the whole and its parts, ceased to function beyond the boundaries of "Finitude".
What if they were both wrong? What if the pasture at the other side only looks greener than our own? What if we are already living the mystery? After all, what would people living in another galaxy think of ours? How wonderful would our world not seem to them? Why should the laws governing Infinity be any different from the laws we know?
In the end, any conception of Infinity that declares it as different from the Finite has to be considered as pure speculation: beyond the Finite, anything goes, and mathematicians can compete about who will draw the most elegant formula. Maybe they could paint it, or put in graffiti form.
[See Hawking 's "Into a Black Hole". I am afraid I do not find his approach very convincing. I find the idea that burning an encyclopedia just makes the information that was contained in it more difficult to decipher quite implausible. But then, it all depends on how you choose to define information I suppose.]

Set Theory: Mathematics or Metaphysics?
Set Diarrhea
The following definition is, as far as I am concerned, written in a very clear and unambiguous style. So, enjoy it while you can.

"AXIOM VII. (Axiom of infinity [Axiom des Unendlichen].) There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as an element." 

This is then how Zermelo defines the (infinite) set of natural numbers in his first axiomatization attempt of 1908:
"[paragraph 14] If Z is an arbitrary set constituted as required by Axiom VII, it is definite for each of its subsets Z1 whether it possesses the same property. For, if a is any element of Z1. it is definite whether {a}, too, is an element of Z1, and all elements a of Z1 that satisfy this condition are the elements of a subset Z'1 for which it is definite whether Z'1 = Z1 or not. Thus all subsets Z1 having the property in question are the elements of a subset T of UZ, and the intersection (No.9) Z0 = DT that corresponds to them is a set constituted in the same way. For, on the one hand, 0 is a common 
element of all elements Z1 of T, and, on the other, if a is a common element of all of these Z1 then {a} is also common to all of them and is thus likewise an element of Z0
Now if Z' is any other set constituted as required by the axiom, there corresponds to it a smallest subset Z'0 having the same property, exactly as Z0 corresponds to Z. 
But now the intersection [Z0, Z'0], which is a common subset of Z and Z', must be constituted in the same way as Z and Z'; and just as, being a subset of Z, it must contain the component Z0, so, as a subset of Z', it must contain the component Z'0. According to Axiom I it then necessarily follows that [Z0, Z'0] = Z0 = Z'0 and that Z0 thus is the common component of all possible sets constituted like Z, even though these need not be elements of a set. The set Z0 contains the elements 0, {O}, {{O}}, and so forth, and it may be called the number sequence, because its elements can take the 
place of the numerals. It is the simplest example of a denumerably infinite set..."

We have therefore the following 14 sets:
Z, Z', Z1, Z'1, Z0, Z'0, UZ, DT, {a}, T, T1, P, PT, intersection [Z0, Z'0]. And that is even without the natural numbers!
I will try to make sense of it by, again, dividing the whole mess in random pieces.
i) "If Z is an arbitrary set constituted as required by Axiom VII [of Infinity]"
We are dealing with an infinite set which can already be considered as the set of all natural numbers. I wonder what all the other sets are good for then.

ii) "it is definite for each of its subsets Z1 whether it possesses the same property."
iii) "For, if a is any element of Z1. it is definite whether {a}, too, is an element of Z1
The first newcomer is a collection of subsets Z1 which we can assume to be equivalent to the whole set Z, in that it contains not only a, but also {a}, and therefore {{a}, and so on.

iv) "and all elements a of Z1 that satisfy this condition are the elements of a subset Z'1 for which it is definite whether Z'1 = Z1 or not."
Down to the next level, or is it up? I must say that I am a little confused here. Are we dealing with a subset of Z1, let us call it Z'1 for fear of being accused of originality, or with a superset of Z1, equivalent to out first set Z? Ah, it is the old trick of subsets being equivalent to the whole set, the identifying property of infinite sets. But if we already know that it is infinite, what are we still doing here?

v) "Thus all subsets Z1 having the property in question are the elements of a subset T of UZ [union of Z]" 
Go back up one level to Z1, and be glad Zermelo in his kindness did not name it Z2 or use some other edifying label. That does not mean we can avoid the fact that our old acquaintances, the subsets Z1, are now considered in their relationship to the union set UZ, of which they are also the subsets. I wonder what that teaches us concerning Z, the set we had started with.

vi) "and the intersection (No.9) Z0 = DT that corresponds to them is a set constituted in the same way." 
That they are also infinite of course! as is the set which is the intersection of all those subsets. How could I miss that! 

vii) "For, on the one hand, 0 is a common element of all elements Z1 of T, and, on the other, if a is a common element of all of these Z1 then {a} is also common to all of them and is thus likewise an element of Z0." 
Where did Zcome from? Ah, yes! from the previous sentence. It is necessarily part of the intersection set with the double name, Z0, and DT, in case we did not get enough of those. Anyway, Z0 contains not only a, but also {a}. Sounds familiar?

viii)" Now if Z' is any other set constituted as required by the axiom, there corresponds to it a smallest subset Z'0 having the same property, exactly as Z0 corresponds to Z." 
Z'... Z'... Didn't we meet this guy already? Wait, no, we did not, it was his brother, Z'1. Z' is the new guy on the block. And he is as powerful as Set Number One, Z. He also has his own sidekick, Z'0, just like Z has Z0. Together, they are going to make sure that everything stays as it should be. Better not mess with them! Infinity is their neighborhood!
I suppose that Zermelo thought that if he bored us to death, we would be more inclined to say he was right just to shut him up. As far as I am concerned, I am ready to agree to anything. So please forgive me if I stop right here!
I still have no idea what paragraph 14 is all about. The use of the principle that infinite sets are equivalent to (some of) their subsets seems to be very premature. Also,If the paragraph is supposed to be a proof,  it is certainly very poorly presented, and as an illustration of the axiom of Infinity it simply sucks. It is though very representative, in a caricatural way, of the tendency of set theorists to create sets more easily than they can fart, if I may be so blunt. It seems like they are so afraid of falling victims to the contradictions, that they prefer to err on the safe side: create a new set at each step, just to be sure that it will not bite you back in ... the long run. I find myself each time having to concentrate intensely on the set creation process before I can even start to make sense of any statement.
And please do not think that Zermelo was the only one doing this.
I think there is something fundamentally wrong with set theory.

Set Theory: Mathematics or Metaphysics?
"Man created the natural numbers, and all other kind of stuff."
Shaito san (as noted by George on the fourth moon of the Year of the Shark).

Well-Ordering Principle and Least Element
Set Theory has spread its tentacles to every corner of Mathematics, even in the venerable Number Theory which is (almost) as old as Mathematics itself. See how Burton ("Elementary Number Theory", 2002) formulates an ancient theorem:

"Theorem 1.1 Archimedean property. If a and b are any positive integers, then  there exists a positive integer n such that na >= b.
Proof. Assume that the statement of the theorem is not true, so that for some a and b, na < b for every positive integer n. Then the set 
S = {b — na | n a positive integer} 
consists entirely of positive integers. By the Well-Ordering Principle, S will possess a least element, say, b — ma. Notice that b — (m + 1)a also lies in S, because S contains all integers of this form. Furthermore, we have 
b — (m + l)a = (b — ma) — a < b — ma 
contrary to the choice of b — ma as the smallest integer in S. This contradiction arose out of our original assumption that the Archimedean property did not hold; hence, this property is proven true."

The theorem is in itself crystal clear. Or so it seems.
If you take two distinct numbers (a and b), and multiply one of these numbers (a) by a third number (c), then the result will be, at least in some cases, either equal or greater than the other number (b). 
Maybe not so clear after all. It sounds like the theorem is saying that the result could be anything! Which is certainly true. It can be less, equal, or greater than b. What kind of theorem is that?!
Look now at the proof: "there exists a positive integer" has suddenly become "for every positive integer"! That is cheating! The theorem never said that the result would always be equal or greater than b, only that it might!

Okay, even if that was the fifth edition of the book (!), maybe Burton somehow overlooked this error, and meant something more like:
Theorem: If a and b are any positive integers, then any positive integer n is such that na >= b."
I still would not know why anyone would want such a theorem, but then, maybe that is why I am not a mathematician.
Let us look now at the proof once again.
Notice the Well-Ordering Principle that stands prominently in the proof. It tells us that any well-ordered set has a least element. How do we know that S is a well-ordered set? Simple, the definition of the principle tells us very clearly: "Every nonempty set S of nonnegative integers contains a least element". Since the elements of S are all positive integers, the case is closed.
But then I start to wonder: what does na mean? Multiplication of a by n, right? so, by definition, na will always be greater than a. We have therefore the following possibilities:
a equal or greater than b: in both cases na will be greater than b, and we would have proved the (new) theorem.
a smaller than b: but how much smaller? Even then, na can turn out to be greater than b. In which case, we would also have proved the theorem.
Please notice that we did not need the Principle of Well ordering (and the existence of a least number). Just like Archimedes also never did. The "nature" of natural numbers was in itself sufficient to bear the weight of the proof.
This clearly shows that, at least in some cases, the principle is completely superfluous. It expresses a property which is inherent to natural numbers in that they would not be natural numbers if they did not possess it. In other words, all the principle says is that natural numbers are... natural numbers.

The idea of a least number sounds very powerful, until we realize that it does not in fact add anything to our understanding of the situation. Dealing with natural numbers, we indeed always know that there will be a least number. Always. In any situation and in any set of natural numbers. If we could use that fact to prove what we want, we would be able to prove anything concerning natural numbers.
Because it is always true it does not tell us anything about the rest of the proof. Even if the proof is invalid, it will still be true that a set of natural numbers has a least element.
The idea of a least element is therefore useless, and in that, it resembles suspiciously the idea of a first element as analyzed above in the entry about von Neumann.
Let us look now at the following point.
"Theorem 1.2 First Principle of Finite Induction. Let S be a set of positive integers with the following properties: 
(a) The integer 1 belongs to S. 
(b) Whenever the integer k is in S, the next integer k + 1 must also be in S. 
Then S is the set of all positive integers."
Here is the proof:
"Let T be the set of all positive integers not in S, and assume that T is nonempty. The Well-Ordering Principle tells us that T possesses a least element, which we denote by a. Because 1 is in S, certainly a > 1, and so 0 < a — 1 < a. The choice of a as the smallest positive integer in T implies that a — 1 is not a member of T, or equivalently that a — 1 belongs to S. By hypothesis, S must also contain (a — 1) + 1 = a, which contradicts the fact that a lies in T. We conclude that the set T is empty and in consequence that S contains all the positive integers."

This is a perfect example of selective thinking. 
k will always be greater than 1, and k+1 greater than k. What else could you possibly need to say that nothing smaller than k could be anything else but 1. That was your assumption to start with. The concept of least element cannot be used as part of the proof. because it is itself the consequence of it.
If any k > 1, than nothing can be smaller than k.
And if it is possible, then there is no reason that some positive numbers smaller than k, could not be elements of T.

Set Theory: Mathematics or Metaphysics?
Well Ordering or the Phenomenology of Numbers
Judith Roitman ,"Introduction to Set Theory", 1990, is very clear where it concerns Well 0rdering:
"The spine of the set-theoretic universe, and the most essential class of objects in the study of set theory, is the class of ordinals. One of the basic properties of an ordinal is that it is a well-ordered set." (chapt.1) She repeats the same idea a few times in the book to make sure the reader gets it.
What she shows though, just like all the other authors I have consulted, is that the concepts of well-ordering, and therefore of ordinals, would be not be possible without the pre-existence of the (natural) number system. The uncanny resemblance of set theory with number theory results is, as far as I can see, much more than a mere coincidence. With this huge distinction that Number Theory has no pretensions when it comes to the origin of its rules and intuitions, while Set Theory would like us to believe that it is explaining the genesis of numbers. While, in fact, it does no more than describe them, just like Number Theory.
And not always in a very convincing manner, witness this simple 
"Example 1. The set of positive natural numbers N+ = {1,2,3,...}, where we define n Div k iff л divides k. Is Div a partial order on N+? "

[Div means "divides", the author uses the less or equal sign with an index D.]
Before we get to the demonstration we need the following definition:

"Here are the axioms defining a partial order s on a set X: 
"For all x, y, z E X, 
PI (Reflexive), x =< x
P2 (Antisymmetric). If x =< у and у =< x then x = у 
P3 (Transitive). If x =< у and у =< x, then x =< z." 

Now we can look at the proof:

"Check for P3: i) If n Div k then k = in for some i. 
ii) If k Div m, then m = jk for some j. 
iii) So if n Div k and k Div m, there are i,j with 
m = jk = jin. 
iv) Hence n Div m."

It looks here like (iii) is a necessary intermediary step to arrive at conclusion (iv). But, what is the difference between them? How do we know (iii) is true?
we have
1) k=in
2) m=jk
And both give us:
3) m=jk=jin.
As far as I am concerned, the jump from (2) to (3) is as big as the jump from (iii) to (iv). So how could one justify the other?
In fact, I find the usual jump, that is, the way it is usually presented without any puha, much more intuitive:
n Div K, K Div m, n Div m. 
They represent both the same logical intuition, which makes (iii) completely superfluous.

Set Theory: Mathematics or Metaphysics?
The arrogance of modern scientists and mathematicians is certainly not entirely unjustified. Their disciplines have proved their worth over and over again. It is therefore not surprising that Galileo's conception of Mathematics as the language of Nature found such a warm welcome in the mind of scientists and mathematicians alike. Kronecker 's statement of faith that "God has created the integers, the rest is the work of Man" may also be understood in this fashion. Numbers are divine tools that, used properly, will unveil all mysteries of creation.
It is certainly difficult, not to say impossible, for a modern philosopher to refute such an optimism. There is certainly no alternative to the scientific approach to natural events, except mysticism and some kind of obscurantism. This showed already in the debate between Al Ghazali and Ibn Rushd in the twelfth century. Ibn Rushd, (Averroes) has been seen as the rational(ist) voice against Oriental mysticism. He was in fact more successful in the West than in his own backyard where Al Ghazzali's views remained predominant.
Still, allow me to break a lance for Al Ghazzali's approach. He never denied the value of science as such, only rejected the thought-imperialism of "philosophers" who were convinced that they could infringe unpunished on divine territory. In modern terms, we would say that Al Ghazzali was a precursor of Gadamar and Habermas in his refusal of applying the same methods and conceptions to physical and non-physical events. It is just regrettable that what the world retained was the negative aspect of his analysis on one side, and his mystical views on the other.
This belief in the undisputed superiority of Science and Mathematics is often expressed unabashedly in this forum like something so evident that it does not need any justification. It is what makes people say "[...] researchers are actual mathematicians with advanced degrees in mathematical sciences, so I think that we do know better.", or "That's why I actually conduct experiments instead of just farting around making up ever-more-elaborate "just so" stories that can only be measured by their internal consistency or how well they fit into some privileged set of game rules (oops, I mean "philosophical paradigm")."
This is also what made an independent researcher, Bhup Anand, whom I asked if he could explain to me without getting too technical, why he thought that the consistency of the axioms of ZFC could not be proven, reply with a condescending remark: "I’m afraid the only way I know of communicating about the issues you want me to explain, without demeaning the dedication of generations of scholars, would require that you accept the discipline needed to first learn from standard texts not only the significance and importance of adequate and unambiguous communication, but also how this is only achievable through a minimum of competence in the language of mathematics."
In other words, Truth is only available to those who speak the language of the Gods.
The problem is of course, and I will not be afraid of repeating myself, that this blind arrogance is not entirely unjustified. Still, while practical sciences are kept in check by their (lack of) results, mathematics in human sciences and pure mathematics can claim anything they want without anyone bold enough to call them to the witness stand. Those who could, have themselves too much to loose, or just do not see the problem for what it is. Which makes Mathematics blind and dangerous.
All my threads have been aiming not at denying the legitimacy of science and Mathematics, for that I am too much of a believer myself, but in rejecting this blind arrogance that goes against the spirit of Science and the Search for Truth, however it may be conceived.

Many mathematicians, and among them Cantor and other illustrious names, have a hard time distinguishing between an axiom, a definition and a theorem. 
Here is a simple illustration of this difficulty, also taken from Roitman (1990).

"Theorem 2. For all x and y, x = у iff (x ⊂ у and у ⊂ x). 
Proof. Given x, y,
x = y iff ∀z(z ∈ x <-> z ∈ у) 
iff ∀z([z ∈ x -> z ∈ у] and [z ∈ y -> z ∈ x]) 
iff [∀z(z ∈ x -> z ∈ у) and ∀z(z ∈ у -> z ∈ x)] 
iff (x ⊂ у and у ⊂ x)." 

The proof can be reformulated in the following statements:
  i) x = y iff any element of x is also an element of y
 ii)          iff any element of x is also an element of y AND any element of y is an element of x
iii)          iff x is a subset of y and y a subset of x.

What kind of proof is that? In fact, except for the last statement, they can all be considered practically identical, even if the syntax differs from one statement to the other.
So, taking (ii) as paradigm, what is the difference between the left and the right side of the equation
(x = y) = (any element of x is also an element of y AND any element of y is an element of x)?

The statement sounds indubitably true to our ears. But what does make it so? Can we really speak of a proof that x = y,  or is it simply the explicitation of what we mean by it? To recognize that an element of x is the same as an element of y, we do not even need the vice versa. When we see that x1 = y1, we immediately see the complementary fact, that y1 = x1. That is not a conclusion since we could not see the first half without the second. It would be like saying that we could draw the conclusion that a is left of b from the fact that b is right of a. Which we of course certainly can. If told the former, we would "deduce" the latter. If Logic is understood as a system of tautologies, then we are rightly entitled to call this a true deduction.
That includes the last step (iii), if understood properly.
Again, the question is whether the preceding statements can be used as premises to what is supposed to be their conclusion. Do they really entitle us to to assert (iii)? Or does (iii) represent an independent thought that stands on its own?
If we try to build a set based on (iii) we quickly realize that if we make the whole of x a subset of y, that is, x a proper subset of y, we could not possibly then turn y into a subset of x. Either every element of y will also be equivalent to an element of x, or some elements of y will remain unclaimed. Understood this way, it would seem that (iii) is the justification of (i) and (ii), instead of the reverse.
Theorem 2 could be said to be the Anti-Cantor-Bernstein Theorem. While CBT assumes the possibility of a set being equivalent to one of its subsets, Theorem 2 only makes sense if that is deemed impossible.

The problem as I see it, is not the legitimacy of Mathematics, but that of the Mathematicians. They, as humans, have no reason to be arrogant. They are as fallible as the rest of us. Which is really sad, because, when you think about it, there is no Mathematics without Mathematicians.


Set Theory: Mathematics or Metaphysics?

Dear Hachem,

If you are referring to our private correspondence (link below), the remonstrance---admittedly and regretfully expressed ‘condescendingly’---is well-intentioned advice that ought not to be dismissed summarily as a subjective slight. El Ouggouti_20071217_20160408.pdf

It is always with an element of sorrow that one sees someone who is as passionate as you are about the topics which you consider, and as gifted in expression, (a) admit to relying on the text of ‘Contributions’ rather than on standard texts; and (b) resist respecting the minimum of competence necessary for a productive dialogue on the issues that you raise.

Kind regards,


Set Theory: Mathematics or Metaphysics?
The Pairing Axiom or the Fallacy of Ordering
Mathematicians have always dreamed of Syntactic Independence for their discipline. The necessity of axioms was recognized early by the Greeks, still, the hope was already there, especially by Pythagoras, that the numbers would one day speak for themselves. This strong desire to liberate Mathematics from the yoke of reality found one of its strongest expressions in Hilbert's axiomatization of Geometry. While Euclid could not go further that reduce concrete objects like points and lines to mere objects of thought without any dimension, Hilbert simply decreed that even in this form, they were not needed anymore. Reality was fired and replaced by Reason. A very impoverished version of Reason as it turned out. Instead of celebrating the supremacy of Thought, the axiomatization movement declared a coup destined to put in power that which had until now been considered as a creation of the mind. If Syntax could be freed of the vagrancies of Mind, then Mathematics would finally come into its birthrights.
Since Hilbert, Church, Post, Turing and von Neumann (to name but a few) happened, and the machines followed. The idea of Mathematics as a huge computer program has been somewhat tempered by Gödel's results, but the hope is still there: to do away with Man and his feeble mind.
This is particularly obvious in what would seem insignificant details of Set Theory, like the following proof:

"Theorem 6. For all sets x, y, z, w, (x, y) = (z, w) iff x = z and у = w. 
Proof. By extensionality and definition 5, if x = z and у = w, then (x, У) = (z, w). 
For the other direction, suppose (x, y) = (z, w). If {x} = {z, w}, then x = w = z and {x, y} = {z}, so x = у = z and у = w. If, on the other hand, {x} = {z} # {z, w}, then x = z # w. Since {x, y} = {z, w}, we again have y= w."

The whole point of this axiom is to ensure that the order of the elements of each pair is unambiguously set. I must admit that I find the whole enterprise quite baffling. After all, it seems to me that a simple definition, or even better, convention would do. If we agree that (x,y)=(z,w) mean that x=z and y=w, then there would be no ambiguity whatsoever. So why do we need more that this simple syntactic convention? And is that even possible?
Apparently, Set Theorists seem to think so, and the above proof can be found in different formulations in any book of set theory.

This is how the proof looks like when taken apart:

0) "For all sets x, y, z, w, (x, y) = (z, w) iff x = z and у = w." 
This is definitely not a syntactic convention but a theorem that has to be proved!

i) "By extensionality and definition 5, if x = z and у = w, then (x, y) = (z, w). "
If we already know that x = z and у = w, then the problem is simple and can be considered as solved by the Axiom of Extensionality (AE). But what if we do not possess such a knowledge, how could we still attain the same conclusion?

ii) "For the other direction, suppose (x, y) = (z, w). "
We want to establish the identity of x and z on one hand, and y and w on the other. We therefore assume that the first part of the formula is true, that (x, y) = (z, w). It seems quite reasonable were it not the fact that we have no way of knowing that beforehand. In other words, when dealing with two pairs, assuming they are equivalent would be quite arbitrary. Still, it makes no sense assuming the contrary, since that would mean the end of the whole endeavor.

iii) "If {x} = {z, w}, then x = w = z and "
This is a simple consequence of the Axiom of Extensionality. If both sets are equivalent, that means that their respective elements are also equivalent.

iv) "{x, y} = {z}, so x = у = z and у = w."
{x, y} = {z} because we already have x=z, and that leaves z as part of the equivalence. All this is only possible, once again, because of AE: {x,x}={x}. As a consequence we can also say 
{x,x}={x,x}={x} and replace each x with a variable of our choice.
This is in fact quite a useless result. By assuming {x} = {z, w} we have turned everything into an x soup.

v) "If, on the other hand, {x} = {z} # {z, w}, then x = z # w. Since {x, y} = {z, w}, we again have y= w."
This the long awaited result. 
But why stop here just because it is what we were looking for?! This is supposed to be a proof, not a scavenger hunt! We should look at the other possibilities!
Suppose then that 
{x} = {w} # {z,w}, 
then we would end up with x = w and y = z, the opposite of what we really want. And there is no way to exclude this possibility on logical grounds. The only way to do that is by convention and the whole proof is a sham! You cannot define order syntactically, somebody has to put it in there first. 

So much for Syntactic Independence.

Set Theory: Mathematics or Metaphysics?
I think that you have made my point better that I ever could. Thank you. 
Regarding my competence, I certainly cannot claim to be a mathematician, but that does not mean that I am unable to look critically at a mathematical proof and examine its validity. What I have done with the Contributions can be done with the other sacred texts you refer too, if not by me personally, then by others better suited for the job. I started with Cantor because that is the absolute starting point as far as set theory is concerned. The fact that you dismiss him so easily makes me wonder how you interpret ZF(C). 
Furthermore, I think my threads show that so-called mathematical proofs are not always what they claim to be.  I will be grateful to you for looking at them critically and show me where I went wrong. I am certainly willing to learn from a master.

Set Theory: Mathematics or Metaphysics?
Numbers and Proofs in Set Theory
Here is an interesting sample presented to us by Derek Goldrei in "Classic Set Theory" (1996).
As the author put it, "We are effectively showing that any rational not in r is an upper bound for r".

"Let r be a Dedekind left set. 
(a) Show that r is bounded above as a subset of ℚ, i.e. there is some x ε ℚ such that q <q x for all q ε r. 
As r is a proper subset of ℚ, there is some x ε ℚ \ r. 
We claim that x is a suitable upper bound of r. 
If this were not the case, there would be some q ε r such that x <q q. But then, as r is closed to the left [...] we would have x ε r, which would give a contradiction." (p.10)

In short, x is an upper bound, because if it were not, it would not be an element of the complement of r, but an element of r itself. Which is a contradiction.
What is wrong with this kind of argument?

To answer this question we have to ask ourselves: what do we want to prove exactly?
In plain words, we want to prove that there is an upper bound. And now the proof tells us that if x is not this bound, then something else will be, plus that that will make of what we thought was the upper bound, an element of r, which contradicts our initial assumption.

Since we are dealing with numbers, the argument that if x is not the upper bound, it will be smaller than another element, will always be true. Either x is the last number in our list, or there will be more numbers in our list. Pretty obvious, right?
Now the second half of the argument: if x is not the upper bound, it has to be an element of r. It kind of reminds me of George W. Bush: "You are either with us, or with the terrorists". In both cases, the universe is very small and clearly defined. Either x is an element of r, or it is not.
Well, I would say that sums it up right there. Once you have established this ground rule, what is there left to prove?
The contradiction attained seems to take on itself the burden of proof, while in fact it is simply the consequence of the rule that has been established before. This kind of arguments is really a cheap way to come by proofs. Make up a rule, then make up an example that is contrary to this rule, and you will have a contradiction.
I know what the obvious objection is: no rule is being made up here, we are talking about rational and real numbers. The rules applicable to these numbers are legitimate mathematical rules.
Agreed. That still does not justify this kind of arguments. If you have established a rule, you are entitled to point at individual cases that violate this rule, and call them a contradiction or whatever names you can think of. What you cannot do is take a violation of your rules, and the contradictions that ensue, as a proof of your rule. There would be no violations without pre-existing rules as any lawyer would tell you.
There are many arguments of this kind in Set Theory, and I think that one important reason is the following:
All rules or theorems concerning numbers have been invented because the mathematicians already had at their disposal the concept of (natural) numbers. The fact that they consciously choose to ignore the thousands of years is has taken Man to invent numbers, does not make it any easier on them. All the so-called proofs they can think of are a product of this history, even more, they cannot be understood without the implicit knowledge of number mathematicians have gained as modern individuals.
In this particular case, the fact that we are dealing with rational numbers is almost irrelevant. What is important is that these rational numbers are just numbers. The whole argumentation is based on this implicit assumption and would be meaningless without it.
It only makes sense because we know, when dealing with numbers, that any number will either be smaller or greater than another number. That is just how numbers are. So, if you think that you can prove that a set has an upper bound and that a number either is that upper bound or not, then you are really stating the obvious. The only thing you can do is prove that a definite number is the upper bound of a definite set. That still leaves a general abstract proof as meaningless, not to say useless.
This "solution" does not prove that x is an upper bound of r, it just shows that if you assume that there is no upper bound to r you will get a contradiction because you initially had assumed the contrary.

If that is what is usually meant with a mathematical proof, then I think I'll pass.

Set Theory: Mathematics or Metaphysics?
I ♥ Contradictions
"Why, sometimes I've believed as many as six impossible things before breakfast!"
Lewis Carroll "Through the Looking-Glass, and What Alice Found There", (1871).

This is what Fraenkel says about the set of all sets that do not belong to themselves.
THEOREM 2. There exists no set (element) which contains exactly those 
.elements which do not contain themselves (in symbols: ¬∃y∀x(x∈y↔x∉x)). 
Proof. By contradiction. Assume that y is a set such that for every element x, xεy if and only if x∉x. For x=y, we have yεy if and only if y∉y. Since, obviously, yεy or y∉y, and, as we saw, each of yεy and y∉y implies the other statement, we have both yεy and y∉y, which is a  contradiction."

Okay. But suppose now x#y. Is there any reason why sets which are not members of themselves could not be members of y? This sounds almost like the Tragedy of Set , only that it is completely different. Except non-somethings, any set would be eligible. And then we get to the crux of the problem. What about y itself? Is it a member of itself, or not? Let us say it is, then y does not belong in itself. Period. There is really no contradiction in such a conclusion. Unless you want it to be. The same can be said in case you decide that y does belong to itself. Also no problem. Here is the proof.

You: Hey, how come you're not in there?
Set : where?
You: in yourself!
Set : are you out of your mind?
You: uh, well, you're not a member of yourself, are you?
Set : so?
You: and all sets that are not members of themselves are in there, right?
Set : so?
You: well, shouldn't you... I mean...
Set : what? I should swallow my tail and then the rest of me? I'm not the Cheshire Cat, you know! Or Ouroboros!
You: who? What?
Set : never mind. See, George said that I am not a basket, that I just point at things. And that is what I am doing right now. I am at the same time pointing at myself, just by being here. It's like when you point at yourself by pressing your index finger on your chest, or your nose, if you're Japanese. There is nothing contradictory in that, is there?
You: uh, no. I don't think so. Oh, heck, what do I care! You will do whatever you want anyway! You would say the same thing if those were all sets that are members of themselves, and you were not.
Set (triumphant): exactly. (to himself) I kind of like this new me!

Set Theory: Mathematics or Metaphysics?
The Axiom of Extensionality and the Fallacy of Intentionality

The Axiom of Extensionality is always presented as the first axiom of Set Theory whoever the author is, while the order of the other axioms differs from one writer to the other.
It sounds deceivingly simple: two sets are equivalent if they have the same elements.
Fraenkel is one of the few authors who does not gloss over the difficulties presented by the concept of identity. In fact, his presentation is somewhat obscure and probably only meaningful for those who have a deep exegetic knowledge of the different versions of Set Theory. ["One of the most fundamental notions of mathematics is the notion of equality." p.25] 

I will therefore limit myself to some general comments as far as this concept is concerned.

"AXIOM (I) OF EXTENSIONALITY. If x⊆y and y⊆x, then x=y; in other words, sets containing the same members are equal. 
In symbols, ∀x∀y[∀z(zεx↔zεy)→x=y]. "

How is that supposed to take place?
We have a set z in mind. It is still empty of any element. Then we think of an element x. It is also only in our mind, because if it already was placed in z it would take a definite shape (a square of a real instead of an arbitrary non-negative number). Then we think of y as an other element of z. At this time we are allowed to think of x as having taken place in set z, but then we realize that having done so, we have at the same time placed y in z. Only, we have not put two elements in z, but only one which we had called one time x, and the other time y.
The question now is, could there have ever been two different elements, or sets, one containing x, the other y?
For that, we have to turn to the distinction between extension and Intension. This is how Fraenkel presents it:
"from an intensional point of view the set of all non-negative real numbers and the set of all squares of real numbers are not necessarily identical, even though they have the same extension." (p.27-28)
Let us assume that the mathematician, somehow, overlooked this identity; in his mind he is dealing with two different sets. Or even better, let us just use variables instead of concrete values or definitions, which make it impossible to know beforehand if two variables have the same extension. Something that must happen everyday in mathematical practice. In fact, the foremost objective of any calculation or theory is to arrive at equations like x=y, whatever the values may be.
In other words, deciding that two variables represent one and the same thing is the Holy Grail of scientific and mathematical research. Who cares then if you decide to put x and y in the same set or not since you will be able from there on to replace both x and y by a common variable? This looks like a very clear example of, as the Dutch would say "mustard after the meal", ["mosterd na de maaltijd"], or too late in the day. [I really like the anecdote of a Dutch minister/state secretary who, when asked by President Kennedy what his hobby was, answered with pride: "I fok horses!" He had to explain to the shocked head of state that he meant breeding the animals, and not subjecting them to intimate and unspeakable acts.]
Anyway, the point is that the Axiom of Extensionality really seems... pointless. The above example of the squares of reals and non-negative numbers make it even more obvious. Are there really two sets, one containing all non-negative numbers, the other containing the squares of reals? I am definitely not talking about the metaphysical significance of the existence of such sets. I will accept any form of existence, no questions asked, even in bitcoins.
What do the words " the set of the squares of real numbers", or "the set of non-negative numbers" mean really? We all agree that they denote the same objects. But what does that entail in mathematical practice? It seems that, each time you are working on one set, you are at the same time working on the other, without you knowing it! And that is the whole point. Once you have discovered their identity you realize that you have been using different names for the same object. At that moment, you decide to rectify your error and use a single variable for both. And consider them as elements of one and the same set.
The fallacy in Fraenkel's argumentation about the primacy of extension above intension is that he is making the same mistake he reproaches the intensional approach: one (and the same object) is considered as many. The equivalence of different descriptions, variables, or set elements, is the final step in a what could be a long and arduous analysis. Once this equivalence has been established, the principle of extensionality becomes trivial. It is what gives meaning to scientific and mathematical research. To state it as an axiom would assume that we have to accept its rule before taking any further step in our research, while it is in fact the implicit objective of any scientific endeavor.
It cannot therefore help us prove anything beforehand, except state useless and trivial facts like: if each time a set has an element x it has an element y, and vice versa, then x=y. 
There can never be an x and a y in the same set unless they are different from each other, or unless we are not aware of the fact that they represent one and the same object. And how would we know that without an intensional judgment? How would we know that the square of a real is a non-negative number unless we understood both expressions as meaning the same? [One can of course have a set of intensional objects!]

This last formulation is by the way different from that of the original axiom. Fraenkel also indicates it by a different token, a star:

"* x=y if and only if every set z which contains one of the elements x and y contains also the other. 
In symbols, ∀z(xεz↔yεz)↔x=y." (p.28) 

The difference between both formulations is quite subtle. In the Axiom, we come to the equality of sets via that of the elements, while in the case of the * principle, the reverse takes place: we conclude to the equality of the elements when they are elements of one and the same set.
Does this distinction make sense in itself? We are relying on the identity of sets or elements to assure ourselves of the identity of respectively elements and sets, and vice versa. It sounds very much like a chicken and egg dilemma.

Fraenkel's delusion consists in believing that he can safely approach extensionality without bothering with Intentionality.