> It seems like it should be possible, in theory, but it takes too much work.
The definition of a valid mathematical proof that I've heard mathematicians use is if it convinces other mathematicians. I think there's integrity in depth in mathematical proofs for lots of reasons, tying back to an axiomatic basis is a lot of extra work for likely no benefit.
On formal mathematics, almost no mathematicians pay attention to this afaik. It's something that lives in philosophy departments, not mathematics.
The definition of a valid mathematical proof that I've heard mathematicians use is if it convinces other mathematicians. I think there's integrity in depth in mathematical proofs for lots of reasons, tying back to an axiomatic basis is a lot of extra work for likely no benefit.
On formal mathematics, almost no mathematicians pay attention to this afaik. It's something that lives in philosophy departments, not mathematics.