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I am looking for philosophical arguments for and against Cantor's theorem other than the ones Cantor came up with, if you know any, can you present them or a link to them?

I post this in philosophy since in the mathematical realm it is a settled issue as mathematics only cares if it is proved, yet philosophy asks why is it true, not just is it or is it not true. If we understood why the argument works, we might be able to answer questions like "why is there no cardinality between the rationals and the reals?"

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    Do you have a source for the claim that "nobody understands Cantor's theorem"? – Eliran Nov 18 '18 at 17:00
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    Cantor's th is a mathematical theorem: we can easily understand it. – Mauro ALLEGRANZA Nov 18 '18 at 17:17
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    @Mauro ALLEGRANZA , Okay you disagree with Wilfrid Hodges who said “But then we come to Cantor’s result, and all intuition fails us. Until Cantor first proved his theorem, nothing like its conclusion was in anybody’s mind’s eye. And even now we accept it because it is proved, not for any other reason.” logic.univie.ac.at/~ykhomski/ST2013/Hodges.pdf last paragraph page 3 – Ivan Hieno Nov 18 '18 at 17:35
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    The proof of the th is very simple; in the finite case, we can list all the subsets of a set A and check that they are "more then" the elements of A. Cantor's argument is general: of course, we can deny the existence of an "actually" (in the Aristotelian sense) infinite set and all the seemingly paradoxical aspects disappear. – Mauro ALLEGRANZA Nov 18 '18 at 17:36
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    Cantor's proof has been used by Russell to derive its own paradox, which has nothing to do with infinity. The issue is with infinity, not with the proof. – Mauro ALLEGRANZA Nov 18 '18 at 17:38
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Cantor's theorem is a prime instance of a diagonalization argument, but far from being the only one. There is nothing particularly mysterious or un-intuitive about diagonalization. I do not know what motivates Hodges to claim otherwise, but he his quite certainly expressing a minority viewpoint here.

Diagonalization does not require any specific mathematical foundations to work, but tends to go through for all of them. There is in particular no real justification for any attempt to resolve conflicts with any pre-existing philosophical baggage regarding infinity by faulting diagonalization.

  • Cantor's theorem is not a diagonalization. You might be thinking of Cantor's diagonal argument. OP's link gave the Wiki page for Cantor's theorem, which says there's no surjection from a set to its powerset. – user4894 Nov 18 '18 at 20:01
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    @user4894 Cantor's theorem is diagonalization, as a special case of Lawvere's fixed point theorem. – Arno Nov 18 '18 at 20:27
  • Ok at that level I agree. But then your answer's suitability as a response to the OP comes into question. What you said is true yet at a very different level than the question hence not a helpful response. Is my point too picky or do you agree? – user4894 Nov 18 '18 at 22:59

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