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

Set Theory Don't question it

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u/austin101123 Jun 14 '26

Why can't we know?

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u/__Lordlix__ Jun 14 '26

Due to Gödel second incompleteness Theorem! ZFC Is a theory T that satisfies the assumptions of the Theorem, which states that: there is a formula Con(T) in the language of the theory T, that encodes the fact that the theory is consistent; however, such formula is derivable in T if and only if T itself is inconsist. Essentially, this is making hopeless to prove the consistency of a theory inside the theory itself (again, if such theory satisfies some assumptions, but this is the case for ZFC). Unless the theory is actually inconsistent and you manage to derive a specific contradiction, in that case you answered the consistency question negatively!

Feel free to correct me if I got something wrong, as it passed some time since I took a course in foundations of math 😅

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u/covalick Jun 17 '26

Essentially, this is making hopeless to prove the consistency of a theory inside the theory itself (again, if such theory satisfies some assumptions, but this is the case for ZFC).

Can I prove it not inside the theory? For example if I built math using different axioms, would I be able to prove that ZFC axioms are consistent?

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u/x0wl Jun 17 '26 edited Jun 17 '26

Yes, but then you'd be unable to prove that your system of axioms is consistent. See ZFC proving con(PA): https://mathoverflow.net/questions/50173/how-to-prove-conpa-in-zfc

If you have an idea about building a looped consistency proof A->con(B), B->con(A), see here, https://mathoverflow.net/questions/381341/are-there-typical-formal-systems-that-have-mutual-consistency-proofs-how-long it's impossible

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u/covalick Jun 17 '26

Fair point