TITLE: Is ZFC not foundation of mathematics?
QUESTION [6 upvotes]: I was reading this answer and was shocked by reading that "There is a possibility that ZFC, the formal system that sounds like it's what mathematicians work in (but isn't really)" I cannot believe that mathematicians do not (necessarily) work in ZFC. 

Crossley, Ash, et al.'s book 'What Is Mathematical Logic? ' says Mathematics=Logic+Set Theory , so I think that first-order logic combined with ZFC creates foundation strong enough for mathematics : is it true? If not what forms foundation of mathematics?
Also, if ZFC is not we are working in, why undergrad texts start with mentioning ZFC?

REPLY [4 votes]: Mathematicians generally work in set theory, yes... But that need not mean ZFC.  In fact, mathematicians speak in second-order language, which is outside the first-order rules of ZFC.  So in that sense, they are outside of ZFC.
For number theory, mathematicians may care that $\mathbb Z$ and $\mathfrak{P}(\mathbb Z)$ exist, but do not care whether 
or not $\mathfrak{P}(\mathfrak{P}(\mathfrak{P}(\mathbb Z)))$ exists.  So in that sense they are using only a small fragment of the sets of ZFC.