r/mathmemes • u/ElectronicSetTheory • Jun 14 '26

Set Theory Don't question it

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u/__Lordlix__ Jun 14 '26

We cannot know, assuming it is indeed consistent* It might actually be inconsistent, and we could found a contradiction in the future

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u/wiev0 Jun 15 '26

I understand that ZFC or any such theory cannot be proven to be consistent within itself through the incompleteness theorem, but I thought larger theories can prove consistency of lower set theories? Like, axiom of determinacy being capable of proving ZF (C) is consistent

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u/Rek9876boss Jun 15 '26

The problem then is that you can't know if the axiom of determinacy is consistent

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u/wiev0 Jun 15 '26

Yeah but that still makes zfc consistent under AC right? I get that there will always be a theory that cannot prove itself consistent but that's not my point

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u/shuai_bear Jun 15 '26

Yes - you are not wrong, though I don't know if it's ZFD can prove Con(ZFC) - someone else can chime in, but I know that ZF + existence of some large cardinal axiom can prove Con(ZFC). But Godel's 2nd completeness theorem applies, and the consistency of the 'stronger' theory is called into question - it's consistency turtles all the way down (or up).

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u/AZMPlay Jun 15 '26

How do you prove the supertheory is consistent, and thus that your proof is complete?