>>13595015>This is a very silly assertion. People know how to do natural arithmetic and prove theorems about it much earlier than they see Peano axiomsI said that Peano axioms are just a formalization of our intuition over naturals; of course people knew arithmetic way before Peano, just like Euler did various proofs that utilize much more complex identities that were proven after his use
>And that definition was quickly discovered to imply contradictions, because it's too general. A more precise definition is needed.No it wasn't, his axioms were too general, but his definition was not. In fact, the whole point for sets were to be a generalized object that could describe everything prior, and it works, on the present everyone describes groups as just a type of sets, manifolds as a type of sets, topological spaces as a type of sets, and so on.
Furthermore, people even are making theories even more general than set theory (e.g., category theories) to describe even more objects.
>>13595022>Now say I wanted to check if my thumb satisfies this axiom. How do I do that? It makes no sense.This is plain ridiculous, is like saying that "say if I want to check if my nose is a number" or "say if I want to check if my car is a vector". Mathematics works with abstract ideas not real objects (though, they can inspire those ideas).
By definition, everything in ZFC is a set, so you don't ask if a thing is a set, you ask if a thing exists or not, because the only way to denote such thing would be in the language of sets.
