[Radim]

> (It kind of begs the question of whether math follows from the axioms we want or axioms follow from the math we want). Plus perhaps some new math would start to unfold as we begin to explore the inconsistent axiom's subtleties.

Only tangentially related, but the same idea comes to mind reading Terence Tao's masterpiece on "Smoothed asymptotics" for divergent infinite sums (e.g. the infamous 1+2+3+4+… = -1/12):

https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri...

Our intuitive interpretation (Σn must be infinite! and surely positive! never -1/12) fails miserably for such infinite series, in the sense that "practical experiments" (QM) hint at reality preferring that bizarro -1/12 interpretation instead. Who is at fault here – our seemingly iron-clad intuition or the experiments? And why the disconnect?

Like you say, what new math unfolds once we accept and internalize this new interpretation and adjust our intuition? Tao's piece offers an excellent basis for that. While we may come up with any interpretations and axioms we like, experiment is the final arbiter on which of these "math worlds" are real.