Almost all fields of pure math use a much stronger set of axioms called ZF and essentially everyone also accepts the axiom of choice (making it ZFC). The axioms in ZF are reasonable but the axiom of choice is surprisingly controversial for an axiom. There are some unintuitive consequences of the axiom but even more unintuitive consequences without it or with the negation of it.

Number theory uses a much smaller set of axioms.

It should be stated that most mathematicians don't really mind the logical foundations of their work when they are actually working in the same way that most programmers don't worry about assembly language or transistors.