[thaumasiotes]

> math now sits on very solid axioms (look up ZFC), they're not questioning that.

People question C all the time. That might be the most prominent ideological difference in mathematical philosophy.

Does it matter? Of course not, but people question it anyway.

[cubefox]

Logicians and philosophers of mathematics have also questioned ZF set theory and "set theory" more generally.

For example the axiom of infinity (by finitists), the power set axiom and first-order theories in general (the downward Löwenheim-Skolem theorem implies that the infinity and power set axioms can't guarantee the existence on an uncountable power set), the fact that ZF doesn't allow a set of everything, and in particular no proper set complements, the fact that the axiom of regularity seems to be useless, etc.

Of course most ordinary mathematicians don't care about all that, because they don't care about ZF(C) or set theory or the foundation of mathematics in general. They rather care about problems in their specific field, like algebraic topology or whatnot.