> currently maths is built on set theory most mathematicians will eventually agree with that when pressed. However, almost none of them know the axioms of ZFC

> currently maths is built on set theory most mathematicians will eventually agree with that when pressed. However, almost none of them know the axioms of ZFC by heart (because they don't need them). You can swap out ZFC for something else and nobody will care very much as long as "the same" (does a lot of work here) theorems, which they know and use it their work, remain true. This is what many theorem provers do. Many aren't based on set theory, for example, but instead on type theory. (You can still do set theory in a framework based on type theory, and vice versa, but the foundations are "different").