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

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?

• Mathematics did live and exist without formal logic for thousands of years. However, in modern times, mathematicians became aware that perhaps we ought to study the fundamentals in order to be certain that mathematics won't collapse under our feet one day. Thus formal logic was born in an attempt to ensure that all our previous results (and future) are indeed sound. Commented Aug 29, 2016 at 14:18

• Thank you very much. Does this mean that if we just argue by using the rules of the axioms given, we are good? I have seen first order logic very complicated with having to define an alphabet etc., but in this article it means that all we have to do is follow those axioms when reasoning? Also there is one thing about the predicate axioms I do not quite understand, they talk about beeing a "member of", $\in$, in axiom 13 and 14, but when working with predicate calculus we do not know if we have sets yet, so how does that make sense? Commented Aug 29, 2016 at 16:44
• @user119615 I don't know what "the case on the site" is, but usually the alphabet would consist of variables (usually x, x', x'', etc.), logical symbols like $\forall$ and $\vee$, set symbols like $\in$ and $=$, and so on. Commented Sep 3, 2016 at 0:39