# Are there any good philosophical arguments for or against Cantor's theorem, other than the ones that Cantor came up with?

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?"

• Do you have a source for the claim that "nobody understands Cantor's theorem"?
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Commented Nov 18, 2018 at 17:00
• Cantor's th is a mathematical theorem: we can easily understand it. Commented Nov 18, 2018 at 17:17
• @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 Commented Nov 18, 2018 at 17:35
• 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
• @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. Commented Nov 18, 2018 at 20:24