2nd order logic and mathematics?

[tgt]

Does most of mathematics use 2nd order logic? If so would studying the foundations of mathematics involve mostly using 2nd order logic?

 

[CRGreathouse]

Studying the foundations of logic usually starts with 0th level and moves to first, then either touches on second and returns to first or 'graduates' to second.

There's value in simplicity, so while many theories can be expressed in second-order logic I generally see them as being embedded in first-order logic. But there's no escaping real complexity; ZF in first-order logic has infinitely many axioms, so perhaps one would argue that its second-order formulation is 'no worse'.