 Back 2016-01-05 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Reference and Existence1) the set of all sets that are not members of themselvesonly poses a problem if 'the set' refers to an object , or whether, as a linguistic element, it functions as a name or description, whether it  is simply used to refer to 'all sets that are not members of themselves". Let us concede that "all sets that are not members of themselves" form a multitude of real existing objects. The question now is whether "the set" is also an existing object, or whether it is simply the name or description of these real existing objects.When we name somebody 'John', we do not create a new entity distinct of John. We cannot say that there is in the world John, and also 'John'.What is peculiar is that the passage from name/description can be made surreptitiously. That is quite evident in:2) The set of all sets that are not members of themselves exists.The ambiguity disappears completely in:3) The set of all sets that are not members of themselves is a misnomer. Permanent link: https://philpapers.org/post/11578 Reply

 2016-01-05 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Separation Angst"Separation: For every set A and every given property, there is a set containing exactly the elements of A that have that property. A property is given by a formula f of the first-order language of set theory." (Stanford Encyclopedia of Philosophy)One property could be: 'is not a member of itself".We would then have a set S whose elements, (belonging already to another set A, distinct from S), would be sets that are not members of themselves.The Axiom of Separation, as the name indicates, creates (an infinite number of) separate sets, one for each conceivable property, out of the original and thus avoids, allegedly, self-reference.The apparent advantage of this axiom above my own analysis is that S can still be considered as a legitimate set, whereas with me it was demoted to a mere name or description.The choice seems quite simple, doesn't it? Let us find out.What is a set that is not a member of itself?A set of dogs is itself not a dog, a set of mountains not a mountain. There are infinite examples of such sets. In fact, most sets will probably be sets that are not members of themselves.Sets that are members of themselves are in fact much more mysterious, and seem to be defined only in a negative way: the set of all non-dogs is itself a non-dog; the set of all non-mountains is itself a non-mountain...Let us look now at the most defining property: elements of S must be elements of A. So if a set is to be considered a set of dogs in S, it will have to be first a set of dogs in A.As it turns out I do not think it is  possible to exclude the inheritance of the antinomy. Let us not forget that A is both a member of itself, and not a member of itself. We will not be able to choose between one of those two contradicting properties, and S will remain at best undefined. We cannot exclude A when it is not a member of itself, and neither can we when it is, because that would mean excluding all members of A, therefore not creating S. Unless you can exclude A without excluding its members? A fascinating possibility!Of course, this objection rests on the assumption of an antinomy within A. But then, without this assumption, why would we need an Axiom of Separation?I would not be surprised if I went wrong somewhere, and if I did, I cannot wait to know where. Permanent link: https://philpapers.org/post/11610 Reply

 2016-01-05 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Reference and geometryWhen Euclid affirmed that we can draw only one line through a point, he had not only understandably failed to anticipate non-Euclidean geometry, he also did not fully realize the meaning of his assertion. After all, how can we draw a triangle, or even a simple angle if both sides cannot not share one and the same point? But they do, don't they? Where is then the problem?Well, in fact that would mean that you can draw at least two lines through a point, which would not be very Euclidean. Unless of course one line is considered as jumping over the point which has already been claimed by its sibling.We know that solid objects cannot do that: you cannot have two different objects occupying the same space at the same time. That is why geometrical lines, just like geometrical points, have no spatial dimensions. Our drawings are mere pedagogical tools to help us understand abstract principles. Furthermore, engineers have learned since Antiquity how to translate those drawings into real constructions.Let us now, for the sake of argument, consider both principles as equivalent:A) you can draw only one line through a pointB) two solid objects cannot occupy the same space at the same time.Suppose that these principles were hard-coded in reality, that is, even drawing more than one line through a single point was physically impossible. How would we draw a triangle?Here is a possible method:1) draw 2 vertical and parallel lines as close to each other as (physically) possible.2) draw 2 oblique lines, each one from the top of the corresponding vertical line with any arbitrary inclination, not necessarily the same on both sides.3) keep drawing oblique lines on both sides, with the same original inclination (s), keeping the same minimal distance with the outer lines as the distance between the vertical lines, starting each time from a lower point respectively on the vertical lines.4) Draw a horizontal line, the base, to form a triangle containing the smaller triangles. You must have now have two large triangles separated by the minimal vertical, or horizontal, distance.5) Draw a few parallel lines to the base between the base and the top.6) you can now see that each line contains a different number of points.This would be a physical refutation of the Cantorian principle that a subset can be put in a one to one correspondence with the larger set.Also, the expression, C) the top of the triangle is a pointwould have to be replaced by: D) the top of a triangle is formed by two points.Let a djinn now suddenly reduce the minimum distance between the points in such a way that it would disappear from our perception. (C) would then become once again a very plausible assertion. But would it describe reality? For that matter, would (D) describe reality?After all, the distance between two points may not be empty. Maybe the following expression would be more appropriate:E) the top of a triangle is formed by three points.But then why stop at three? Will CERN help us one day find the right expression?This shows how hazardous it is to let language refer when it is obvious that even its users have no idea what they are talking about. Permanent link: https://philpapers.org/post/11726 Reply

 2016-01-05 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Mradulla PatelLSPFar Eastern Vaishnav This is inspiring. But logic and phenomenologic claim proof of truths and falsehood,can also be found in between 'statements' after the modern greats in this area of thought from the 2nd WW. Say Heidegger's dasein, even Kant's noumena are full experiments in the scientific philosophical sense. Ethics is the underlying driving force to make this possible. So Pegasus is an example of this with no accomodation in the material sense but a truth in the phenomenological. This states that logic must be perceived beyond the mind in psychology to make it vast and open to philosophers. Permanent link: https://philpapers.org/post/11730 Reply

 2016-01-05 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Remark on Reference and Geometry:I realize now that the Euclidean geometric rule concerns the drawing of one line through two points. Still, I do not think that it makes any difference for my argumentation. After all, when drawing a triangle you have to draw two different lines through each point. Permanent link: https://philpapers.org/post/11738 Reply

 2016-01-05 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti A Djinn in ParadiseImagine you are a powerful djinn who caught all natural numbers in a magic bucket. You want now to enumerate different objects to see what all the Cantorian commotion is about. You are especially interested in the different forms of infinity since you find the one in which you have been restrained all your life somewhat... restraining. The idea of a greater freedom is a powerful motivation, and you get to work.You decide to investigate the infinitely small regions first, that would put that much more distance between you and the Big Boss, giving you more opportunity to expand your horizons. You follow Cantor's instructions and start putting all the reals (at least the part after the decimal point) between 0 and 1 in an infinite list, making use of the ingenious diagonalization method of this smart human.After an infinite while you stop and ponder. It does seem like there can come no end to the list of reals between 0 and 1. Would it be possible for a canny djinn to hide between those reals after a major mischief?You look at the infinite list for a long, infinite time until you finally notice the bucket from which you have been pulling all those numbers to draw those reals. You sit and ponder.What would happen to the reals if this magic bucket ran empty? What if the Big Boss took it away while you were hiding in there? How would you ever come out of it? Permanent link: https://philpapers.org/post/11742 Reply

 2016-01-05 The Liar Paradox (and other beasties) Reply to Priyedarshi Jetli I hope my other entries answer somehow to your first objection that I did not deal with a formal paradox.As far as Habermas is concerned, he is certainly not the only one, nor the first to speak of natural language as its own metalanguage. Concerning his conception of truth you might look up his work on communication, starting with his monograph published in a book with Niklas Luhmann in 1971 "Theorie der Gesellschaft oder Sozialtechnologie". Permanent link: https://philpapers.org/post/11838 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Addendum to Dedekind's cut:Can we account for all the points of a diagonal in a unity square?Let us look at the method described above (the entry Reference and Geometry).There is no reason to start at the top. We can draw parallel lines to the side starting from the base.But before that, we have to make sure we are dealing with a square and its diagonal. By drawing the second diagonal we create four equal  triangles. The point on each half of a diagonal can be put on a one to one correspondence with the  points of the other sides respectively, taking into account the principle of one line for each point. This puts Dedekind's remarks that some points cannot be put in one-to-one correspondence with rational numbers in a new light: the only explanation is that the points only detectable with irrational numbers are in fact not located on the line itself, but are a product of his own formulas.Anyway, a very interesting question is, looking ata2 < D < ( a+ 1)2[I have replaced the lambda sign with a]and knowing that "this cut is produced by no rational number", what does this formula refer to? Permanent link: https://philpapers.org/post/12074 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Pythagoras and Perspective[Ye be warned who enter here: the following offers a pure view of the mind, unhindered and unspoiled by any trivial knowledge of worldly affairs.]The fact that the diagonal of a unity square is itself different from 1 has posed quite a conundrum not only to philosophers and mathematicians, but also to science in general. How come we are unable to measure exactly the diagonal using the same unit as with the sides?Even irrational numbers are not really a satisfying solution, except in a practical sense. After all, the diagonal has a fixed, precise and delimited length which can be measured using other unities of length.It seems like the distance between two points differs with the perspective. But why should it surprise us? After all atoms are not perfectly circular, and if we can go by their big brothers, the stars and planets, they generally tend to the elliptical, as do their orbits. The axial distance of a heavenly object will therefore differ according to the perspective of the observer.Understood this way, what would be surprising is if there were no difference at all between measures taken from one perspective and those taken from another. Permanent link: https://philpapers.org/post/12106 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Where are my 2.3 kids?Statistics is obviously a domain that will always make use of irrational numbers. There is therefore hope for the other numbers too. The metaphysical implications will still disappear from mathematical statements. Nobody believes seriously in 2.3 kids. We all know it is, like  Gauss would say, a mere façon de parler. Permanent link: https://philpapers.org/post/12126 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (2)"We will call by the name "power" or "cardinal number " of M the general concept which, by meansof our active faculty of thought, arises from the aggregate M when we make abstraction of thenature of its various elements m and of the order in which they are given."Cardinal number is here a concept, and not a set.This become less obvious with the following lines:"Since every single element m, if we abstract from its nature, becomes a "unit," the cardinal number M is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate M."Here a cardinal number starts looking like the set itself, or the object of which it is supposed to represent the number of elements.The jump becomes a giant leap when Cantor declares:"between the elements of M and the different units of its cardinal number M a reciprocally univocal (or bi-univocal) relation of correspondence subsists."It sounds like we do not have a single number anymore, but as many as there are members in a set. And that brings us to the mysterious statement (9):"For, as we saw, M grows, so to speak, out of [cardinal] M [with two upper bars] in such a way that from every element m of N a special unit of M arises. Thus we can say that              =(9) M ~ M." [the second M is also cardinal M, with two upper bars.]Understood this way, each element or unit of the set becomes a unit of the cardinal number, and therefore '1'. Every set is a set of 1's, which makes it easy to calculate the final cardinal numbers, and the results of all known (arithmetic) operations. This certainly seems to make sense since we had stripped the elements of a set from any specific property other than their membership of the set. We then might as well call them 'units' or '1's', both expressions being equivalent.The problem of course is the concept of a cardinal number on one hand, and the individual units or 1's which, when added up (if the operation in question is addition), yield that cardinal number.It seems to me that, even if we consider the cardinal number as a set, we have in one case a set with only one number, the cardinal number, and in the other case a set containing different units or 1's (which, taken together, represent said cardinal member).(9) is therefore very problematic, to say the least. Permanent link: https://philpapers.org/post/12138 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Corrections to Cantor's Logic (1)The text editor interprets the "greater than" and "less than" signs as the start of special computer instructions, like TAB or END OF LINE (slash n).This resulted in truncated sentences.[I use italics for the sentences as they appeared and bold fonts for the corrections.]Error#1"Here again, condition (a) is sufficient for expressing N>m (or M" should  end with "(or M less than M)" [you understand if I do not use the symbol itself.]Error#2"We are compelled to draw the same conclusion as before: (a) is a very round-about way of saying that M"should be"We are compelled to draw the same conclusion as before: (a) is a very round-about way of saying that M Less Than N (or N Greater Than M). But instead of presenting the fact on a platter, we make the hearer (or reader) work for it."[as you see, the rest of the paragraph following the special symbol was deleted!]Error#3"In other words, any way we look at it, M(b) is also a self-sufficient condition for the > and < relationship."are in fact two sentences:- "In other words, any way we look at it, M less than N (or N greater than M)."The following sentence is supposed to stand alone:- "(b) is also a self-sufficient condition for the > and < relationship." Permanent link: https://philpapers.org/post/12146 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor' Logic (3): The Addition and Multiplication of PowersThe identification of the cardinal number with the elements of a set pays its dividends right away. Like I mentioned before, it makes the introduction of arithmetic (and other mathematical) operations seem quite natural.For instance the sum of two sets poses no problem at all. One only needs to remember that the cardinal numbers M-double bar and N-double bar are equivalent to the numbers a and b.And as we know:a+b=b+aIf we add a number c we geta+(b+c)= (a+b)+c.These rules, and others similar to these, seem self-explanatory. Until we ask ourselves which cardinal numbers we are speaking about.When we say a+b, do we mean the cardinal number denoted by a single number, like 5 or 7, or the different units making up this cardinal number?Take the schema we used before:M________________ Nxx_______________ xxyx________________ xxx_______________ xxyHow do we add M to N without the use of numerals, be they that of natural or cardinal numbers, and whatever the difference might be?I honestly would not know, at least not where large numbers are concerned, and I do not think Cantor did either!Natural numbers seem to surreptitiously hitch a ride by every definition of mathematical operations on sets. Permanent link: https://philpapers.org/post/12150 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Eray OzkuralBilkent University It's easy to deal with antinomies in set theory. The axiom of infinity is false. Much ado about nothing. Permanent link: https://philpapers.org/post/12166 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor-Bernstein Theorem: a real or apparent problem?There are many proofs of this theorem, that Cantor called the Equivalence Proposition (Satz), but of which he never published a proof. In fact, Arie Hinkis' book "Proofs of the Cantor-Bernstein Theorem" (2013), is more than 400 pages long, and he does not even cover all of them. So you will understand if I choose the simplest one of them. To be honest, the most proofs I would not even understand!I found Borel's version ("Leçons sur la Theorie des Fonctions", appendix I, 1898), of Berstein's proof reasonably readable (that goes also for Kertesz' presentation in his "Einfuhrung in Die Transfinite Algebra", 1900) but if push comes to shove, I think I will work with my own version!As you know, the whole point is to prove that two sets, A and B, are equivalent when they are known to be equivalent to one subset of the other, respectively A' and B'.A~B' AND B~A'--> A~BOne very obvious method is to enumerate the different possibilities (there are 4 of them) concerning the relationship between a set and its corresponding subset. Then it appears that only one possibility is of interest for us, the case where, as Borel put it, A has the same power as B', and B as A'.Hinkis (p.60 and further) is quite critical of Borel and finds the enumeration of the different possibilities superfluous. I personally think it rather sets the argument in the right perspective.  It certainly helped me in rephrasing the theorem, hopefully without betraying its essence. 1) If a subset A' of one set A is equivalent to a set B2) and a subset B' of the set B is equivalent to the set A3) then the two sets A and B are themselves equivalent.Here again we are confronted with what I would call the Oracle-style or the Tongue of Riddles. The obscure formulation creates, in my eyes, the false impression of a problem that has to be solved, the empty conviction of the necessity of a solution that has to be proven logically and/or mathematically.But what happens if we solve the linguistic riddle first?10) if one part of A, including the whole of A, is equal to the whole of B,20) and if one part of B, including the whole of B, is equal to the whole of A,30) then the whole of A is equal to the whole of B.(10-30) can be simplified even further:11) If the whole of A is equal to the whole of B,21) and if the whole of B is equal to the whole of A31) then the whole of A is equal to the whole of B.And now I wonder: what have all those proofs been proving for more than hundred years?[Hinkis: "The naive reaction of someone educated in college set theory, is that CBT for sets of infinite numbers must be trivial. Our proof shows that this is not the case in the context of early Cantorian set theory." (p.20)That may be so, but the way the theorem can be rephrased is not refuted by considerations of infinite sets and their peculiar relation to their own subsets. In fact, assuming that subsets of infinite sets have the same power as their parents makes the simplification of the theorem (from 10-30 to 11-31) even more acceptable. This makes Hinkis' speculation that Schroeder committed suicide because he was ashamed of his mistake even more tragic!] Permanent link: https://philpapers.org/post/12198 Reply

 2016-01-12 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor' Logic (4): The Exponentiation of PowersAfter having secured addition and multiplication, Cantor turns to a crucial part in his analysis, the creation of ever larger sets via exponentiation of powers. He starts by defining the concept of function in a set."The element of M bound up with n is, in a way, a one-valued function of n, and may be denoted by f(n); is is called a "covering function of n". The corresponding covering of N will be called f(N)."Cantor calls f(n) a "covering function of N with M".What is worth our attention is the identification of the element m with the function that connects it to a corresponding element n.The cautious "in a way" should be taken very seriously.The following lines create their own problem though."The totality of different coverings of N with M forms a definite aggregate with the elements f(N) ;we call it the "covering-aggregate (Belegungsmenge) of N with M " and denote it by (N | M).(2) (N|M) = {f(N)}."What is meant here is the totality of f(N)'s, and that is a to the b'th power, ab, with a and b, as before, being the cardinal number of respectively N and M.The question is, does this totality even exist (in the mathematical or logical sense)? After all, once you have defined two sets by a function, you have once and for all determined the nature and properties of the elements of both sets.Let us look at a simple example.M____f(N)____ N4____ f(2)____ 29____ f(3)____ 325___ f(5)____ 5I wonder if it would be possible to define a function completely different from "the square of n" for those two sets. So, in a way, ab  only makes sense if we consider the elements of f(N) as indeterminate units which can be represented by any function at the same time.Still, that would demand an extra factor, the number of possible functions, which we do not have. No, what is meant here appears to be  a pure numerical relationship between N and M. Each element of N is assumed to have M possible values. But that is quite arbitrary, isn't it?This is obviously a case where the identification of a set with its cardinal number is very problematic. ab may make sense mathematically, but it lacks any logical consistency.And I am not even stopping to wonder whether f(N) can be legitimately considered as a set. After all, it looks very much like the name of the relation between N and M: "M is the square of N". Dividing this statement in its different occurrences:"4 is the square of 2; 9 is the square of 3;25 is the square of 5",does not (necessarily) turn this relation into a set distinct from N and M. But then, in the so-called "Naive Set Theory", anything can be a set. I should remark however that even though such a set does not, as far as I can see, create any antinomy, its legitimacy remains questionable.When we consider that exponentiation of powers is the introduction to fundamental propositions concerning transfinite numbers, then it becomes more important than ever to evaluate this step as critically as possible. Permanent link: https://philpapers.org/post/12202 Reply

 2016-01-15 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti The Pitfalls of Infinity: Dedekind's cutIn "Essays on the Theory of  Numbers" Dedekind wants to prove the continuity of the straight line but he finds himself limited by the nature of the Pythagorean dilemma: there is no common unity between the diagonal and the sides of a unity square. There is therefore, according to him, no possibility of accounting for all the points in a line using rational numbers. The only solution to save continuity, is by introducing irrational numbers. "Since [...] it can be easily shown that there are infinitely many lengths which are incommensurable with the unit of length, we may affirm: The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals."This affirmation has apparently a great convincing potential since it has never been put in doubt, even though it is far from obvious. After all, a straight line is only incommensurable with other lines under certain conditions. Any line will be commensurable at least with itself. It is therefore not evident, nor proven, that "The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals."Apparently Dedekind has based all his argumentation on a wrong assumption. I wonder what that means for his analysis?The reader will understand that as a non-mathematician I need to be overly cautious in anything I can say on this subject. I will certainly take my time. That is the least I can do.Anyway, until now, neither physical arguments (Physics assumes discrete quanta of matter) nor geometrical considerations (the gaps in a straight line are not necessarily present) seem to support his approach. Permanent link: https://philpapers.org/post/12070 Reply

 2016-01-15 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti "Thank you for the fish"I am sure you have seen a documentary of whales, dolphins or other sea hunters working together to encircle a school of smaller fish.Imagine yourself a lone hunter/fisher, armed with a long net, but this time you are not hunting sardines but atoms and other imponderables.Let us assume that each point on the perimeter you want circled, where the prey is waiting in fear, is an atom, or any other physical unit. You wouldn't need to worry about formulas for surfaces any more, neither make use of the venerable method of exhaustion, nor appeal to modern integral calculus. All you would need to do is draw a line between each pair of opposite points, respecting the one line per point principle (lets us call it the Point-Line-Point or PLP, to avoid embarrassing alliterations). The length of each line will also immediately indicate the number of units, and calculating the surface of an irregular plane, or its volume, will be as easy as adding the values of all the lines together, even after taking into account the ways atoms are placed to avoid the Pythagorean dilemma [see above Pythagoras and Perspective].A future scanner should be able to give accurate measures of any geometrical figure, without having to worry about decimal points or rounding figures.Sigh, I wish Geometry was that easy in high school. But then, knowing the teachers, they would have made us learn the boring stuff anyway. Permanent link: https://philpapers.org/post/12274 Reply

 2016-01-15 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (5.1): The Finite Cardinal NumbersAfter having given us a taste of what we can expect with the exponentiation of powers, Cantor judges the time ripe to lay down the foundations of his number systems, starting with the most elementary numbers.The number "1":"To a single thing e0, if we subsume it under theconcept of an aggregate E0 = (e0), corresponds ascardinal number what we call "one" and denote by1 ; we have            =(1) 1 = E0 ." [cardinal number with two upper bars]This is a very interesting definition of '1", the mother [or father? It does look like a phallic symbol] of all unities.First, we have "a single thing" which everybody would just call '1", or 'one thing'. Everybody that is but Cantor.Second, we need a (concept of a) set to put that single thing into.Third, once we have that set, and the thing in it, we are allowed to call the cardinal number of that set '1'.What was the cardinal number of a set again? Allow me to quote myself from the first part of this study:"each element or unit of the set becomes a unit of the cardinal number, and therefore '1'. Every set is a set of 1's".I must admit that the use of '1' was an extrapolation from my side. Cantor did not actually use the expression. Still, how are we to understand the statements he made a little bit earlier on in the text?"Of fundamental importance is the theorem that two aggregates M and N have the same cardinal number if, and only if, they are equivalent : thus,                                   =  =(7) from M~N we get M=N,and         =  =(8) from M=N we get M~N."How could a cardinal number be anything else but a natural number applied to (the elements of) a set?Are we really dealing with a new concept, or merely with the technical designation of a familiar concept? If you are speaking of objects in general you may use the expression 'natural numbers', but when dealing with sets and their elements, we prefer you to use the technical term 'cardinal number'?Or could it be that I am underestimating Cantor's geniality, that I am missing a distinction so subtle, but oh! So fundamental between the concept of a cardinal number and that of a mere (natural) number?We can after all see that two small sets are equivalent, and for larger ones, we can always use the one-to-one correspondence to establish which one is greater or less than the other. Wasn't that what I myself showed in the first part where neither the concept or expression of natural number, and for that matter, nor the concept of cardinal number was used to explain the "Greater Than" and "Less Than" relationship?Such primitive notions as the so-called "subitizing" or the one-to-one-correspondence method may certainly be considered as the forerunners of the concept of number, they could hardly replace it. The concept of "cardinal number" is much too advanced to be anything like those primordial beginnings.It remains very strange though that the concept of natural number is considered to be born out of those same atavistic schema's that in Cantor's analysis gave rise to that of cardinal number.A last possibility, as far as I can see, would be that cardinal numbers, unlike, natural numbers, still have to earn their numerals, or titles so to speak. The fact that both kinds of numbers make use of the same numerals would be nothing more but a matter of expediency and economy of symbols.The problem with such a view is that it consecrates the distinction between the two groups without explaining it. In fact, it legitimizes the distinction, making us falsely believe that we already know of its existence and that we are okay with it.We see here the same m.o as in the previous crime scenes: Cantor gives the impression that he is defining something new, while in fact he does no more than reformulate in his own obscure words what was already clearly known.What we have to grant him is that he did not explicitly use '1' until now. Nevertheless, we can hardly say that the use of the concept or numeral '1' comes as a surprise.This will become even more evident with the definition of 2, 3 and the rest of the numbers, whatever their nationality.The number "2""Let us now unite with E0 another thing e1, andcall the union-aggregate E1, so that ,(2) E1 =(E0, e1) = (e0, e1).The cardinal number of E1 is called "two" and isdenoted by 2 :            =      (3) 2 = E1."Okay then, we have 1, as element of a set E0, and we somehow bind it with another thing. Without being too pedantic let's try to do justice to Cantor's intentions and choose a charitable interpretation whenever possible.(2) states that the union of a set with an object [(E0, e1)], is the same as the union of two objects [(e0, e1)].In other words, we do not have to bother with the distinction between set and object at all since any combination will give us the same cardinal number.I am bound to be charitable, but I still need to point to the ambiguity of (e0, e1) and choose the most favorable interpretation: (e0, e1) can be understood as simply a (spatial) arrangement of two objects, or it can be seen as a set which is the result of the union of two objects.In the first case, speaking of cardinal number would be inappropriate since this kind of numbers has only been defined in the context of sets. When dealing with a group of objects, natural numbers seem the way to go.We will consider Cantor justified in this case in his use of the concept of cardinal numbers and we should not give to much attention to the mixed usage of objects and sets, especially since later models of Set Theory will generally abandon the concept of object altogether and refer exclusively to sets.Still, the main problem remains: how do I understand the distinction between a cardinal and a natural 2? If 2, or should I say '2', is to be more than a numeral that just happens to look and sound as the familiar '2', then there must be, once again, a way to distinguish one from the other. And if there is not, my question, as to the meaningfulness of the concept of cardinal number altogether, would be entirely justified.The question becomes even more crucial when we realize that without the concept of natural number the passage from 1 to 2 would remain inexplicable.After all, the only thing that the cardinal approach can show for its efforts are two isolated facts: the naming of a certain set as '1', and another as '2'.In fact, the more cardinal numbers Cantor creates, the more urgent it becomes to assume a "number sense" (as Dehaene would say) which would run implicitly through the whole creation process. When summing up this process by"E2 = (E1, e2), E3 = (E2, e3),...,"and then declaring :"The use which we here make of these numbers as suffixes is justified by the fact that a number is only used as a suffix when it has been defined as a cardinal number."Cantor is asserting something he certainly did not prove: the meaningfulness of the concept of cardinal number, and its right to supplant the concept of natural numbers is far from evident, not to say utterly unjustified.That is particularly obvious when we look closely at the following definition:       =    ="(6) Ev=Ev-1+1;""+1" (or -1), is a concept that is unaccounted for in the definition of cardinal numbers. It belongs exclusively to the number sense. Without the latter (6) would be no more than a name or description and the appendix '+1' would be a literal expression without any numerical meaning.In fact, at each step of the creation process of cardinal numbers, Cantor takes the next natural number implicitly and sneaks it in to "create" yet another cardinal number.It may be not so flagrant in the case of small numbers as in the expression already mentioned:"E2 = (E1, e2)"It becomes more than obvious if we let E2 be equivalent to a large set whose elements could only be counted to make sense of the expression. However we look at it, Cantor gives us no reason to consider cardinal numbers as in anyway different from natural numbers. The whole process looks like a superfluous, and historically probably unrealistic, duplication of the genesis of numbers.Let us hope he will be able to make sense of them later on. Permanent link: https://philpapers.org/post/12334 Reply

 2016-01-15 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (5.3): The Finite Cardinal Numbers (continued)e does not belong to N"(b) The part N of (M, e) does not contain e as element, so that N is either M or a part of M. In the law of correspondence between (M, e) and N, which lies at the basis of our supposition, to the element e of the former let the element f of the latter correspond. Let N = (M1, f) ; then the aggregate M is put in a reciprocally univocal relation with M1. But M1 is a part of N and hence of M. So here too M would be equivalent to one of its parts, and this is contrary to the supposition."This looks suspiciously like a repetition of moves. The same curious argument is used as by the condition that e belongs to N, or that N=(M1, e). There Cantor used whatever implicit rules he trusted on, namely what was to become the axioms of Well-Ordering and that of Choice. That allowed him to consider, against all odds, M1 as equivalent to M. He uses in this second part the same strategy.First, he reformulates N, to which e does not belong anymore, toN=(M1, f); f being the equivalent of e in (m, e).Having done that, he simply assumes M~M1, and declares it as a contradiction, and therefore as a proof that (M, e) is not equal to any of its parts.The fact that M1 could not possibly be equivalent to M remains apparently unseen or ignored.Why M1 cannot be equivalent to M?M1 is a part of N which is itself a part of M. N can be therefore at most equivalent to M, but since that is not allowed, it will always be less than M.M1 is necessarily less than N, since N=(M1, e). A final remarks that holds for the whole of the proof of D. Cantor starts with (M,e) but considers in his conclusions only M, ignoring e altogether. We must not forget that the aim was to prove that (M,e), when M was finite (not equivalent to any of its parts), was itself still finite. I have the strong impression that Cantor himself got confused by his conception of "part of". Instead of considering (M,e) as a totality, he slips and starts looking at the M of (M, e) as the new totality. That would explain his formulation "here too M would be equivalent to one of its parts, and this is contrary to the supposition".He should have first assigned P=(M,e) as the new value of M; then tried to show that P could not possibly be equivalent to one of its parts.As it is, he has been comparing not (M,e) or P to N and M1, but M to N and M1. The problem is that there is no reason why N could not be equivalent to M, or to P-e. And since M1 itself could then be equivalent to N, it could also be equivalent to M, but this time without contradiction.In my own analysis I have tried to take into account Cantor's original intention, that of proving that the whole of (M,e), therefore P, is equal to none of its parts. So, however we look at it, Cantor has not proven that (M,e), just like M, is not equal to one of its parts, because he got confused and considered M itself both as a whole [as in M], and as a part [as in (M,e)]. He forgot his own formulation of D in which he set to prove that "the aggregate (M,e)... has the same property of being equal with none of its parts." (M,e), not M alone which was already assumed as not being equal to any of its parts.The reason for this confusion lies, I think, in the use of e or an equivalent of it in all formulas. Because of it the distinction between a part of (M,e) and a part of M was lost. Permanent link: https://philpapers.org/post/12366 Reply

 2016-01-19 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Diagonalization and its problemsIs there a difference between actual infinity in one dimension, and on multiple dimensions?When we consider the set of even numbers (or uneven, rational...) as countable we are assuming two things.1) We can use the set of counting numbers in an orderly way, without having to turn back on our tracks.2) infinity can be actualLet us take a closer look at the second assumption, and take all reals, starting from 1 (we can always get back to those starting with zero later on). Is is possible to conceive, independently of the question of countability, of those numbers as an actual infinity? If it is then we have an infinite series of infinite series which can be all considered as actual. After all, it would be very difficult to justify an exception rule for natural numbers since each one of those infinite series is itself made out of natural numbers.We can consider all those actually infinite series of reals as forming a two-dimensional matrix, but that would be for our convenience only. There is no reason the number of dimensions could not itself be infinite.Take now any one of those actually infinite series, it will be, as already stated, in principle countable. Let an infinite number of djinns each put one series in a one to one correspondence with the set of natural numbers.This infinite number of djinns will then then be able to count all reals, won't they?[Are you worried about the Axiom of choice, how would each each djinn know which series to take? Let us say that each djinn just picks the first series close at hands and looks for a corner where to work in quiet. There may be some duplicates of series, but who cares? The djinns have their numbers working for them too.]Does your head hurt? Maybe that is because Cantor was right also. Humans cannot count the reals the way they can count even numbers. That does not mean that so-called uncountable sets are in themselves uncountable. Their uncountabilility is a property of our knowledge, not of their nature. Permanent link: https://philpapers.org/post/12514 Reply

 2016-01-20 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Eray OzkuralBilkent University Mathematical induction works, actually. The AGI (abbrv. for Artificial General Intelligence, aka general-purpose machine learning) system I developed did solve such problems in 2010, and there are many experiments by others. We can apparently induce both functions and mathematical formulae.The final system will be able to induce much better than (limited) human mathematicians can. Carnap's theory has been vindicated, and we actually are using pretty much the same theory as Carnap proposed. IOW, I think that you are wrong on all accounts, but this is not something to worry about. We are refuting Platonism wholesale, refuting the vapid criticisms of induction like the so-called "problem of induction" or the white raven paradox, was just the first salvo. Of course, AGI researchers are actual mathematicians with advanced degrees in mathematical sciences, so I think that we do know better. Permanent link: https://philpapers.org/post/12570 Reply

 2016-02-09 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (14.3) Fundamental Series (p.157 ff)Let us look first at the result Cantor has obtained in the previous lines:(F is beta in the text)21) F1 = a1; Fv+1= av+1-av .The first part, F1=a1, seems simple enough. The second part not so.Remember our example?F={1,2,3,4,5,6}, [set of ordinal numbers]G={1,3,6,10,15,21}. [set of sums of ordinal numbers]let v=2Fv+1=F3=3av=a2=G2=3av+1=a3=G3=6av+1-av=6-3=3=Fv+1The series a1,a2,...av, with each following element greater than the one preceding it, will be called "fundamental series".Properties of a fundamental series"(a) The number F is greater than av for every v, because the aggregate (G1, G2,. . ., Gv), whose ordinal number is av, is a segment of the aggregate G which has the ordinal number F"This is a logical trap that I found very difficult to avoid. I kept taking concrete example for F and G, and found each time the opposite of what Cantor was claiming: F < G, just like (21) showed. Here is a "wrong" example as it was used, rightly, before:F={1,2,3,4,5,6}, [set of ordinal numbers]G={1,3,6,10,15,21}. [set of sums of ordinal numbers]Imagine that we are now dealing with infinite sets, then we would have to speak ofFv={1,2,3,4,5,6,...}, Gv={1,3,6,10,15,21,...}. In this perspective, any v will be smaller than the whole infinite set F, and any set Gv will be smaller than G.Cantor's obscure style gives a false impression of depth.The second property of a fundamental series sounds much more straightforward:"(b) If F' is any ordinal number less than F, then, from a certain v onwards, we always haveav > F'."But cantor would not be Cantor if he kept things simple.This is his short proof:"For, since F' < F, there is a segment B' of the aggregate G which is of type F'. The element of G which determines this segment must belong to one of the parts Gv ; we will call this part GV0. But then B' is also a segment of (G1,G2,. . ., Gv0), and consequently F' < av0. Thusav > F'for v >= v0."_________________________________________ F__________________________ F'                          Gv0_________________________________________ G__________________________ B'Looking at the drawing, we realize how ridiculous Cantor's argumentation is. Gv0 is defined as the point that determines the segment B', or as what I called Cantor's cut. Since av0=Gv0, all the points beyond Cantor's cut will have a higher value. This a matter of definition. In fact, a linguistic analysis of (b) would have shown that a proof, even if possible, was completely superfluous:"(b) If F' is any ordinal number less than F, then, from a certain v onwards, we always haveav > F'."(b') If F' is any ordinal number less than F, from a certain v onwards, that is from any v greater than F', then av > F'."So, how do you prove a tautology? You make it sound very complicated.The Concept of LimitThis is, I think, what Cantor has been aiming at all along. He had to give the impression of a final destination that kept shifting away and always just out of reach. And I must admit, he certainly succeeded in his plan: instead of starting from the "obvious", a point in infinity that keeps moving away, he gets to it by successive "proofs" that seem to finally bring the unreachable infinite within our grasp. There is nothing wrong with the rhetorical objective, except that it is wrapped in a pseudo-mathematical blanket.Here is the long awaited statement:"Thus F is the ordinal number which follows next in order of magnitude after all the numbers av ; accordingly we will call it the "limit" (Grenze) of the numbers av for increasing v and denote it by Limv av..." (p.158)I think it is better for me to stop right here. Limit is an essential mathematical concept and it would be pretentious of me to try and discuss it as I did the other concepts Cantor has used to get to this point. As I said before, the fact that Cantor botched the job of founding this concept on a secure footing does not mean that the concept of Limit should be thrown overboard. Even I know that it is much too useful for that. Whether there is a better foundation for it is an epistemological matter that only mathematicians can research. I will leave it to them. Permanent link: https://philpapers.org/post/13522 Reply

 2016-02-09 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (15.0)The Voyage beyond InfinityAt the end of section 14 Cantor starts defining the concept of Limit. As I said, I do not intend to discuss the validity of this concept as such, nor its mathematical use. Still, Cantor uses the Limit concept to justify the further construction of his system. I must therefore look critically as his presentation of this concept, and the proofs he puts forward to justify the existence of other transfinite numbers like Aleph One. For that, I will start at the end of section 14 and move from there to section 15 and beyond."To every fundamental series {av} of ordinal numbers belongs an ordinal number Limv av which V follows next, in order of magnitude, after all the numbers av ; it is represented by the formula (22)." (p.158)(22) Limv av= a1 + (a2-a1) + . . . + (av+1-av) + . . .This is of course a direct consequence of formula (21) which I have analyzed in the previous entry.Let me first remark that (22), whatever its practical utility in mathematics, is not a number, transfinite or otherwise. It is more of a construction procedure which gives a definite number as a result, once v has been defined. The number constructed will by definition be a finite number.In contrast to (22), the following two statements seem to concern definite numbers:(23) Limv (y + av) = y + Limv av, ;(24) Limv y . av=y . Limv av .But here also, unsurprisingly since (23) and (24) are theorems, we are dealing more with a schema than with definite numbers. Both statements tell us how to handle specific numbers having the forms stated.There are no formulas for infinite numbers because that would be a contradictio in terminis. Any formula would necessarily create a definite number, which can therefore only be finite.Cantor does not seem to be aware of this difficulty. In fact, his optimism is such that he thinks he can prove the existence of a stratified Infinite. The first step is of course the distinction between the finite and the infinite. A distinction which he has dealt with many times. He does it again here, trying for a last time to prove this crucial distinction.Finite ordinal numbers and their proof"two different ordinal numbers a and b cannot belong to the same finite cardinal number v." (p.159)Again, given the definitions of ordinal and cardinal, an obvious statement that Cantor sets out to prove.The above statement can be reformulated as follows:Two sets with a different number of elements, and different order of precedence, cannot have the same number of elements.How do you prove that? Cantor knows just what to do:"For if, say, a < b and Go = b, then, as we know, there exists a segment B of G such that Bo = a. Thus the aggregate G and its part B would have the same finite cardinal number v. But this, by theorem C of section 6, is impossible. Thus the finite ordinal numbers coincide in their properties with the finite cardinal numbers."i) For if, say, a < b and Go = b, then, as we know, ii) there exists a segment B of G such that Bo = a. iii) Thus the aggregate G and its part B would have the same finite cardinal number v. But this, by theorem C of section 6, is impossible. Thus the finite ordinal numbers coincide in their properties with the finite cardinal numbers."Where does the conclusion of (iii) come from? It is in fact a reductio. All that rests Cantor is then to show that a set of finite numbers and one of its parts cannot share the same cardinal number. And since C of section 6 says that no finite set can be equal to one of its part, the problem is solved.Cantor's logic can be summed up as follows:1) create an extra element, or level2) declare the problem obviously solved for that element or level,3) declare the whole problem solved.The creation of an intermediary step, of segment B, seems here to make the proof possible. But segment B only emphasizes the fact that a and b are different, and that therefore either a smaller than b, or vice versa. The logic which makes us decide that a part of a set and the set itself cannot have the same cardinal number, is only acceptable as proof if it also applies to two different and distinct sets. And if it does, then we do not need the extra intermediary step with segment B.Cantor has therefore not proven that "finite ordinal numbers coincide in their properties with the finite cardinal numbers". It is either obvious, and therefore in no need of a proof, or it is still unproven.This is more important than some will maybe realize. If this simple fact cannot be proven, then it becomes even more difficult to prove that infinite sets can be equal to one of their parts, and that different infinite ordinal numbers can belong to the same cardinal number, which is anything but obvious!And that is exactly what Cantor wants us to believe:"to one and the same transfinite cardinal number a belong an infinity of ordinal numbers which form a unitary and connected system." (p.159)Cantor calls this system "number class Z(a)", or "the first number-class".Cantor is ready to start the construction of his famous edifice: different levels of Infinity. Permanent link: https://philpapers.org/post/13546 Reply

 2016-02-09 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (15.1) The Numbers of the Second Number-Class Z(aleph zero )There are many proofs in this section too, so we will deal with them one by one to avoid being overwhelmed by Cantor's logic.Proof of A"A. The second number-class has a least number w=Limv v.Proof. By a> we understand the type of the well-ordered aggregate (1) FO = (f1,f2, ...,fv). where v runs through all finite ordinal numbers and(2) fv -< fv+1,Therefore ( section 7)(3) w=F0o, [omega = ordinal F0]and ( section 6)(4) wo= aleph zero [remember, w is an ordinal number, so ordinal omega is a cardinal number]Thus w is a number of the second number-class, and indeed the least. For if y is any ordinal number less than w, it must ( section 14) be the type of a segment of F0 . But F0 has only segmentsA = (f1,f2,...,fv),with finite ordinal number v. Thus y = v. Therefore there are no transfinite ordinal numbers which are less than w, and thus w is the least of them. By the definition of Limv av given in section 14, we obviously have w=Limv v." (p.160)We learn something essential from this proof: Cantor has nothing new to offer.A and its proof are in fact a mere recapitulation of what Cantor is convinced he has achieved.1) v represents all finite ordinal numbers;2) w is the ordinal number of v;3) aleph zero is the cardinal number of v;4) Limv v is what comes after v;5) what comes after v is the end point of all finite ordinal numbers, and the start of transfinite numbers, therefore w as ordinal number, and the same as cardinal number, or aleph zero.This last statement might sound a little bit strange. We would expect the transfinite numbers to start with aleph zero +1, or Z0+1.But remember that Z0+1=Z0, just like w+1=w.Limv v is therefore the highest number possible of all finite numbers [which as limit is never achieved], and the lowest of the next class, the Second Number-Class.Why is the fact that A does not contain anything new so essential? Because it shows that in last instance Z0+1=Z0 is indeed the cornerstone of the Cantorian edifice, as I stated in 6.0. In fact, once the invalidity of that statement was established, all that has come afterwards was in a way superfluous. From a false theory anything can be proved, so that even if Cantor had somehow presented formally valid proofs, they would have still been considered as illegitimate. His failure to do so will weigh even more heavily in the final evaluation of his achievements: he not only based his system on a false principle, he could not even present formally valid proofs in most of the cases. Permanent link: https://philpapers.org/post/13550 Reply

 2016-02-09 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (15.2) The Numbers of the Second Number-Class Z(aleph zero ) (continued)Proof of B"B. If a is any number of the second number-class, the number a+1 follows it as the next greater number of the same number-class.Proof. Let F be a well-ordered aggregate of the type a and of the cardinal number Z0 :(5) Fo = a,(6) a0 = Z0.We have, where by g is understood a new element, (7) a+1=(F,g)o.Since F is a segment of (F, g), we have(8) a+1 > a.We also have ordinal (a+1)=ao+1=Z0+1=Z0 (section 6).Therefore the number a+1 belongs to the second number-class. Between 0 and a+1 there are no ordinal numbers; for every number y which is less than a+1 corresponds, as type, to a segment of (F, g), and such a segment can only be either F or a segment of F. Therefore y is either equal to or less than a." (p.161)While A could be understood as a recapitulation of all past steps, B can be considered as the confirmation of the start of a new infinite series beyond all finite ordinal numbers.We must therefore look beyond the triviality of a+1 as the follower of a, and see that, in Cantor's mind, it gives substance to the abstract idea of the second number-class.The proof is, for a change, quite straightforward, and is summed up in "ordinal a+1=ao+1=Z0+1=Z0". It shows that we have left the plane of finite ordinal numbers, and definitely entered the realm of transfinite numbers, with w and Z0 as the door to this realm.The problem of course is that this proof, like all others, is based on the validity of Aleph Zero.Proof of C"C. If a1,a2,. . ., av,. . . is any fundamental series of numbers of the first or second number-class, then the number Limv av (section 14) following them next in order of magnitude belongs to the second number class.Proof. By section 14 there results from the fundamental series {av} the number Limv av if we set up another series b1,b2,...,bv,..., whereb1=a1, b2=a2-a1,...,bv+1=av+1-av,...If, then, G1, G2, . . ., Gv ,are well-ordered aggregates such thatordinal Gv = bv,then alsoG=(G1,G2, ...,Gv,.. .)is a well-ordered aggregate andLimv av = Go.It only remains to prove thatGc=Z0Since the numbers {b1,b2,,...,bv,,... belong to the first or second number-class, we havecardinal Gv =<  Z0. [=< less or equal]and thuscardinal G =< Z0.Z0=Z0But, in any case, G is a transfinite aggregate, and so the case Gc < Z0 is excluded.We will call two fundamental series {av } and {a'v } of numbers of the first or second number-class (section 10) "coherent," in signs:(9) {av}ll{a'v},if for every v there are finite numbers l0 [lambda 0] and m0such that(10) a'l >av, l >= l0,and(11) am v, m >= m0." [m is the Greek mu in the text]Normally I would skip this proof because it looks so much like the previous one. But the expression "It only remains to prove that Gc=Z0" drew my attention. It sounds so promising, doesn't it? Unfortunately, Cantor doe not have much to show for it.Here is what his proof consists of: 1) Gc is either equal or less than Z0.2) we have established that that means that it is also equal to Z0.Z0.3) Gc is therefore a transfinite number and can therefore never be less than Z0.Nothing unexpected therefore.Proof of D concerns the coherence relation between two limit series, and is, I think, a pure mathematical matter of which I could not say anything meaningful. I will therefore skip it.The following proof is much more interesting.Proof of E"E. If a is any number of the second number-class and v0 any finite ordinal number, we have v0+a=a, and consequently also a-v0=a.Proof. We will first of all convince ourselves of the correctness of the theorem when a=w. We have       _________w= (f1,f2,...,fv...),        __________v0= (g1,g2,...,gv0),and consequently              ___________________v0+w= (g1,g2,...,gv0,f1,f2,...,fv...)=wBut if a > w, we havea=w+(a-w),v0+a=(v0+w)+(a-w)=w+(a-w)=a." (p.163-164)Theorem E is in itself not really surprising. It confirms an essential property of the Cantorian transfinites: any arithmetical operation on a transfinite number (addition, subtraction, multiplication), will leave that number unchanged. The proof does not deviate from the beaten paths, still, one point deserves our attention. The last three lines of the proof are:i) But if a > w, we haveii) a=w+(a-w),iii) v0+a=(v0+w)+(a-w)=w+(a-w)=a.The question is, what kind of number is a-w? Is it a finite or transfinite number? And how could a transfinite number be greater or less than another transfinite number when anything we add to it or subtract from it gives us the same number anyway? [more on that later]If a can be greater than w, than (iii) should not be possible. If it is possible, that means that a cannot be greater than w, and then we may rightly wonder what the whole proof is about.As you can see, Cantorian logic has to bite its own derrière.You cannot on one hand claim that transfinite numbers remain unchanged, and on the other hand, suspend that rule whenever it suits you.We may not therefore, consider E as proven. Permanent link: https://philpapers.org/post/13562 Reply

 2016-02-09 The Liar Paradox (and other beasties) Reply to Hachem El Ouggouti Cantor's Logic (15.3) The Numbers of the Second Number-Class Z(aleph zero ) (continued)Proof of F"F. If v0 is any finite ordinal number, we have v0.w=w.Proof. In order to obtain an aggregate of the type v0.w we have to substitute for the single elements fv of the aggregate (f1,f2,...,fv....) aggregates (gv,1, v gv,2,. . . , gv,v0) of the type V0. We thus obtain the aggregate(g1,1,g1,2,...,g1,v0,g2,1,...,g2,v0,...,gv,1,gv,2,...,gv,v0),which is obviously similar to the aggregate {fv }.Consequentlyv0.w=w.The same result is obtained more shortly as follows.By (24) of section 14 we have, since w=Limv v,v0w=Limv v0v.On the other hand,{v0v}||{v},and consequentlyLimv v0v= Limv v=w;so thatv0w=w." (p.164)Cantor is still establishing the validity, read legitimacy, of all arithmetic operations on ordinal numbers of the second number-class. Namely, the fact that the combination of finite and transfinite numbers leave the latter unchanged. The use of Limits as an alternative proof method is supposed to reinforce this legitimacy.The problem is that the coherence of two limit series does not bring any extra layer of truth. Once you have accepted Cantor's assumption of aleph zero and his different number-classes it makes perfect sense that the limit of an infinite series will also remain the same, whatever you do with the original ordinal number; as long as it is adding, subtracting, or multiplying a finite and a transfinite number together.Things change when both parts of the operation concern numbers of the second number-class, like it is the case in the following proof with a and w.A new MysteryProof of G"G. We have always(a+v0)w=aw,where a is a number of the second number-class and v0 a number of the first number-class.Proof. We haveLimv v = w.By (24) of section 14 we have, consequently,(a+v0)w = Limv(a+ V0)v.But [the preceding subscripts are a tally of the recurring expression](a+v0)=1(a+v0)+2(a+v0)+...+v(a+v0)       =a+ 1(v0+a)+2(v0+a)+...,+v(v0+a)      =1a+2a+...+va+v0      =av+v0.Now we have, as is easy to see,{av+v0} || {av},and consequentlyLimv (a+v0)v=Limv (av+v0)=Limv av=aw." (p.165)I would beg the reader to examine my analysis of this point even more critically than can be expected because I hardly believe it myself!In this proof Cantor makes of course use of the same assumptions as in other proofs. His treatment of v0 is therefore wholly understandable within his framework. But the fact that he assumes that it does not apply to a is simply staggering. I would except a statement like thisi) a.w=w,but instead I getii) a.w=awIn other words, when dealing with numbers of the second number-class, Euclidean logic is magically restored![book IV of the Elements, "Theory of Proportion"]How is that possible? I might have missed something of course, but I could not find any explanation in the preceding sections. Why the normal rules of arithmetic suddenly become valid again when we are dealing only with numbers of second number-class is something I cannot, as of now, explain. Permanent link: https://philpapers.org/post/13566 Reply