 Back 2016-02-16 Set Theory: Mathematics or Metaphysics? Preliminary ObservationSet 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 PrincipleThe 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. Permanent link: https://philpapers.org/post/13662 Reply

 2016-02-16 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Humor?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 yii) (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 withiii) => x E ABut 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 y2) AND IF y belongs to A3) 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? Permanent link: https://philpapers.org/post/13674 Reply

 2016-02-16 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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 like0,{0},{{0}},{{{0}}},...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. Permanent link: https://philpapers.org/post/13682 Reply

 2016-02-16 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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. Permanent link: https://philpapers.org/post/13698 Reply

 2016-02-16 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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 ______________a___________________ M x1,x2,... 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. Permanent link: https://philpapers.org/post/13702 Reply

 2016-02-22 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Carlo LoriCambridge University 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.  Permanent link: https://philpapers.org/post/13806 Reply

 2016-02-22 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Carlo LoriCambridge University Also re your first observation: the claim is about the primitives of mathematics. Permanent link: https://philpapers.org/post/13810 Reply

 2016-02-27 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. Permanent link: https://philpapers.org/post/13834 Reply

 2016-02-27 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti The chicken and the egg: a story without endOne 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. Permanent link: https://philpapers.org/post/13874 Reply

 2016-03-07 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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? Permanent link: https://philpapers.org/post/13934 Reply

 2016-03-07 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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+x5) 9x=96) x=1Such 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 =1ii) 10x=9.999... Lim 10x=10iii) 10x= Lim 9+xThis 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.thereforeiv) 9x = Lim 9 is false, even though bothv)x= Lim 1 andvi) .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. Permanent link: https://philpapers.org/post/13966 Reply

 2016-03-07 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Fraenkel Goes to Hallewood"Theorem 4. A finite set is not equivalent to any proper subset of itself."(p.38)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 {s1, s2, . . ., 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:S={s1,s2,...,sn,sn+1}S'={s1,s2,...,sn,x}x#sn+1Notice 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'."S'=S"+xSo, S" does not rate a mystery element y. It is plain S'-x.To be consequent, Fraenkel should have saidS"={s1,s2,...,sn,y}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}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}S'={s1,s2,...,sn,sn+1,y}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}S'={s1,s2,...,sn,sn+1,rock}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.  Permanent link: https://philpapers.org/post/14014 Reply

 2016-03-08 Set Theory: Mathematics or Metaphysics? Reply to Peter Eggenberger 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. Permanent link: https://philpapers.org/post/14042 Reply

 2016-03-08 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti En Attendant GödelGödel'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 then3) 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. Permanent link: https://philpapers.org/post/14054 Reply

 2016-03-08 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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…. Permanent link: https://philpapers.org/post/14058 Reply

 2016-03-08 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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?  Permanent link: https://philpapers.org/post/14074 Reply

 2016-03-11 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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."Proof.  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.x1_y1y2_x2x3y3We want: (I have made the link longer to indicate that it is two ways)x1__y1y2__x2x3__y3"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.] Permanent link: https://philpapers.org/post/14102 Reply

 2016-03-17 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Wolfgang CernochPostgreaduate society in Vienna, Austria 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 Permanent link: https://philpapers.org/post/14178 Reply

 2016-03-17 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Wolfgang CernochPostgreaduate society in Vienna, Austria 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. Permanent link: https://philpapers.org/post/14190 Reply

 2016-03-17 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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! Permanent link: https://philpapers.org/post/14198 Reply

 2016-03-17 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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. Permanent link: https://philpapers.org/post/14226 Reply

 2016-03-17 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Cantor-Bernstein Theorem (CBT) viewed by HessenbergMy 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! Permanent link: https://philpapers.org/post/14230 Reply

 2016-03-21 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti 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 thereforeM(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. Permanent link: https://philpapers.org/post/14242 Reply

 2016-03-21 Set Theory: Mathematics or Metaphysics? Reply to Wolfgang Cernoch 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. Permanent link: https://philpapers.org/post/14254 Reply

 2016-03-29 Set Theory: Mathematics or Metaphysics? Reply to Hachem El Ouggouti Well Ordering or the Phenomenology of NumbersJudith 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 =< xP2 (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 have1) k=in2) m=jkAnd 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. Permanent link: https://philpapers.org/post/14510 Reply