>>12752642Math isn't deducible to inference rules of inference as generally math isn't about manipulating axioms, for if it were, people would've come up with a human-surpassing theorem prover.
Model theory is the "axiom playground" approach to math, and it barely offers anything to other fields of math, despite being rather sophisticated.
I feel math isn't grounded on axioms or any formalism at all, it's as if axioms are there for convenience to guide us and play a similar role as notation does. Like for example ZFC can turn out to be inconsistent, we don't know if it isn't, making half of our theorems "incorrect" in the general context. But surely that contradiction inside ZFC would be too complicated that it'd bother only a few set theorists. Just like before ZFC mathematicians used inconsistent set theories and the "set of all sets" thing didn't bother them.
Even the concept of proof is subjective, like many proofs from Italian algebraic geometry school wouldn't be accepted as proofs today. Nevertheless their results are still valid. So formalism can merely be a window through which we observe math.
It's only locally where we can make math seem neatly logical, bigger picture always slips away.
>DESU, that's actually a good question I don't have an answer for off the top of my headThe bizzare thing about axiom of choice is that both its acceptance and denial infer non constructive statements which go against our intuition. For example if the axiom of choice is false, then there's an infinite set of real numbers without a countably infinite subset. Meanwhile its acceptance gives us non definable entities like the Vitali set or things like Banach Tarski Paradox. Maybe those things can be made more valid outside of possible domain of human thought.
So as long as you accept the law of excluded middle, you accept those esoteric intangible objects.
