I'm sure AI could contribute to this, but this is already a well-developed field of mathematics, and most of the consequences of additional axioms have been worked out. (The most productive hypothesis has been what's called "projective determinacy", if you're curious.)
replyMathematicians have also gone in the opposite direction, and tried to work out what are the weakest foundations where different results hold. This is called "reverse mathematics".
What name does this "well-developed field of mathematics" go by? (I just want to get a taste of what the field is like.)
replyI also thought that there are an infinite set of possible extra axioms, e.g. axiomize any statement that's true but not provably so via Gödel's First Incompleteness Theorem, though maybe the vast majority of such axioms are "uninteresting".