Can ZFC be taken as the foundation of mathematics without first-order logic?

Question

On the wikipedia page for ZFC: link

It says that ZFC is formally a theory in logic. And I guess all of mathematics relies on logic, but we use logic informally when creating mathematical arguments.(This seems to be done in all textbooks.)

Does this mean that mathematics can live or exist without formal logic? As have been said on this site mathematics existed before formal logic. So is it correct to say that we can build up analysis, algebra etc., with ZFC and logical arguments, but not using formal first order logic?

Answer 1

Norman Megill addresses this here: http://us.metamath.org/mpeuni/mmset.html#axioms

Note. Books sometimes make statements like "(essentially) all of mathematics can be derived from the ZFC axioms." This should not be taken literally—there's not much you can do with those 7 axioms by themselves! The authors are assuming that you will combine the ZFC axioms with logic (that is, the axioms and rules of propositional and predicate calculus). Between ZFC axioms and logic there is a total of 20 axioms and 2 rules in our system.

If you start working with these it quickly becomes clear that you need the logic axioms to do much of anything.