I have never been formally trained in logic and philosophy. I became increasingly interested in the foundation of mathematics after I graduated from university.
Recently, I've been self-studying ZFC set theory and have realized that mathematical reasoning requires propositional logic, which is even more fundamental than set theory itself. I came upon with the so-called first order logic from this wikipedia page. It says that first order logic is the standard formalization Peano axioms of arithmetic and ZFC axioms of set theory, which I believe is the foundation of mathematics.
But after reading a few paragraphs of the "Syntax" section, I became very confused. It seems to me that lots of the definitions in formation rules in first order logic need the concept of set and natural numbers in the first place. Doesn't this seem like a loop?
Maybe one can make some compromise by using the term "collection" (in the plain English sense) instead of using the word set and hoping that there's a consensus on what collections mean, but still one cannot avert using integers.
Maybe I should rephrase my question in another way: what exactly is the relation between first order logic and ZFC axioms? Which one is more fundamental?