Not happy to respond to LLM talk, but you seem interested anyway. Some sleight of hand happens between "fixing a formal system" and using Cantor's theorem for t

Not happy to respond to LLM talk, but you seem interested anyway. Some sleight of hand happens between "fixing a formal system" and using Cantor's theorem for the metamathematical analysis, as if we use classical set theory anyway. Note that you cannot construct any particular example of a non-definable set, which should cast doubt of existence. I'll disagree by pointing to anti-classical set theories. The axiom of infinity proves independence from ZFC, so I can freely replace the axiom of infinity with its negation, then the natural numbers no longer form a set. Some constructive analysis systems include an axiom that every real-valued function is continuous (as discontinuous functions are undecidable). https://en.wikipedia.org/wiki/Axiom_of_infinity#Independence https://en.wikipedia.org/wiki/Axiom_of_infinity#Independence https://en.wikipedia.org/wiki/Constructive_analysis#Anti-classical_schools https://en.wikipedia.org/wiki/Constructive_analysis#Anti-cla...