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Now set theory is taught even to kids and it is the foundation of mathematics. Can we say that Cantor won?

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    $\begingroup$ Yes and no: Constructive Mathematics is a well developed field of study. $\endgroup$ – Mauro ALLEGRANZA May 11 '16 at 19:28
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    $\begingroup$ The answer is yes, and you yourself explained why. Constructive mathematics is something few specialists study while set theory is taught to everyone. $\endgroup$ – Alexandre Eremenko May 11 '16 at 19:43
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    $\begingroup$ Constructive mathematics is also taught to everyone, just under the name of computer science. When it comes to ideas "winning" makes little sense in the fullness of time, Aristotle "won" for far longer than he "lost" afterwards, so what? $\endgroup$ – Conifold May 12 '16 at 4:49
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    $\begingroup$ The substantial parts of set theory are not taught to children, but the concepts they need in order to start on that later are sometimes taught to children. $\endgroup$ – Michael Hardy May 12 '16 at 19:27
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    $\begingroup$ This question is fairly ill-posed; you might want to specify what you mean by "Cantor won". $\endgroup$ – Jonathan Cast Aug 17 '16 at 15:14
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No, Cantor did not "win", for a very simple reason: the race is not over. Cantor may be in the lead, but there is no reason to think that Kronecker ( or somebody else ) will not be in the lead 100 or 200 years from now. Also, it is incorrect to say that X is the foundation for mathematics. There are multiple competitors for that title, and the best bet ( IMO) is that there is no one true "foundation" for mathematics.

P.S. The major competitors today are Category Theory and (Homotopy) Type Theory. There are probably others, but those two are pretty well-known.

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Very much so. Today someone arguing Kronecker's position would be regarded as a crank (sort of like a finitist). Kronecker was arguing for way more than just "constructive mathematics"; he believed things like:

  • There is not set containing all the subsets of an infinite set (like the integers or the rationals).
  • The nested interval property (that all infinite sequences of nested closed intervals have a common point) may fail (even in constructive math, this property holds for sequences given by an algorithm).
  • Consequently, he believed that irrational and transcendental numbers do not exist (there are constructive proofs that numbers like $\pi$ and $e$ exist, and any classical proof whose result is a negation is also valid in constructive logic, so they are transcendental).
  • For similar reasons, Kronecker's argument would imply that there is no set of real numbers (even if individual irrational or transcendental numbers were granted to exist).
  • And, in general, Kronecker would deny the existence of uncountable seets (Brouwer, for example, took for granted that the real numbers were uncountable).

So there is a long list of propositions Kronecker held to that modern constructivists don't believe; they might criticize aspects of Cantor's set theory but not the whole concept of infinite sets the way Kronecker did.

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  • $\begingroup$ Brouwer took for granted that "Cantor's second number class does not exist."[L.E.J. Brouwer: "Over de grondslagen der wiskunde" Thesis, Univ. Amsterdam (1907) pp. 5 & 144] $\endgroup$ – Otto Jun 4 '17 at 16:52
  • $\begingroup$ are finitists considered cranks? i met Bishop who said that integers were not infinite but continuing (or something like that) -- is that considered a fringey position? $\endgroup$ – releseabe yesterday
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Set theory was already around before Cantor used it to theorise his notion of cardinals. It's a mistake then to associate his name solely with set theory, and personally I think he, himself would be horrified to be so honoured. Like almost all great mathematicians, he knew how much he owed to both his colleagues and his predeccessors.

It's also a mistake to think of mathematics as a competition or a race. To do so is to turn it into a kind of parlour game, which it emphatically is not. More politically speaking it's symptomatic of neoliberalism - a horizon under which science, as a whole, does not succeed as it should - which perhaps is the point of neoliberal disciplinary systems.

So the answer to your question is - it's the wrong question.

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Kronecker won, Cantor lost. Plenty of material proving this case is given in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

EDIT

In Chapter V Mückenheim has collected a tremendous amount of over 250 critical voices, most of which are directly refusing set theory or its basic assumption, namely actual or completed infinity.

Further he applies mathematical limit theory of analysis in order to show that not all rational numbers can be mapped on natural numbers. The fact that every rational can be mapped on its own natural is irrelevant, he says, because every rational number belongs to a finite initial segment which is followed by potentially infinitely many rationals. The mathematical limit of the not mapped rationals is infinite. He gives convincing evidence and a formal proof which to my knowledge has never been refuted. So if set theory is taken as it is, it is in contradiction with analysis and therefore cannot be its foundation.

Then he has shown that an irrational number can never be defined by its digit expansion because there is no number defined unless the last digit is known. If only a string of digits is given - without a finite formula like "0.111..." defining the sequence - then no real number is fixed. Therefore Cantor's diagonal argument does not yield a fixed real number let alone a transcendental number. The finite formulas like those known for $e$ or $\pi$ however are countable. By the way, even Hessenberg, a strong supporter of set theory and author of the famous uncountability proof using all subsets of the natural numbers, has recognized that "a statement about the number defined by that process, which concerned only that number, would be possible only after completion of the process – and this process cannot be completed."

Finally, Mückenheim shows that, if countability of set theory is assumed (contrary to its being in contradicition with analysis), all objects of thought can be enumerated by their rational spatio-temporal coordinates even in an infinite and eternal universe. Therefore there is nothing uncountable.

There are many other arguments. But I would recommend to read first the statements in chapter V. They show that by far not all mathematicians agree to Cantor's victory over Kronecker.

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  • $\begingroup$ @Roy Daulton: Please see the edit. $\endgroup$ – Otto Jun 5 '17 at 10:33
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    $\begingroup$ Thanks for the additional information. I note that your initial claim is that "Kronicker won, Cantor lost," a very strong claim, but the evidence you provide shows only that a relatively small number or 250 out of the 80,000 or so mathematicians disagree with Cantor. Your evidence does not really support your claim. $\endgroup$ – Rory Daulton Jun 5 '17 at 11:01
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    $\begingroup$ @Roy Daulton: First: There is mathematical evidence given that has never been refuted. Second: The taste of the majority is not the best taste. This holds always and not only in arts. Third: I guess that of the 80000 or so mathematicians 75000 do not even know the axioms of ZFC set theory and never have heard of counter arguments. Ask someone of your acquaintance what s/he thinks of "real" numbers which can never be defined and never can be applied in mathematics. $\endgroup$ – Otto Jun 5 '17 at 12:32

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