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"?
    – E...
    Commented Nov 18, 2018 at 17:00
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    Cantor's th is a mathematical theorem: we can easily understand it. Commented Nov 18, 2018 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
    Commented Nov 18, 2018 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. Commented Nov 18, 2018 at 17:36
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    @IvanHieno I read your linked PDF. I now understand what's tripping you up. Here's a way to break down "for no other reason". The author is effectively positing that there are two reasons people choose to believe something: either it is "obvious"/"common sense"/"intuitive", or it's been proven. Here, when he says "because it's been proven, and for no other reason", the implication is "Cantor's theorem is true, we know this because it was proven true, not because it's immediately obvious or intuitive". Hope this helps.
    – Dan Bron
    Commented Nov 18, 2018 at 20:24

1 Answer 1


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
    Commented Nov 18, 2018 at 20:01
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    @user4894 Cantor's theorem is diagonalization, as a special case of Lawvere's fixed point theorem.
    – Arno
    Commented Nov 18, 2018 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
    Commented Nov 18, 2018 at 22:59

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