Is FOL a formal system?

Question

If a formal system consists of a formal language and a deductive system, and that a deductive system consists of inference rules and axioms, then why do we say that FOL is a family of formal systems since (from what I've heard) FOL doesn't have any axioms? So I then assumed that FOL + some axioms is a formal system, but then what is FOL alone?

Answer 1

We can set up systems of FOL which have logical axioms as well as rules of inference. Indeed, this is the traditional way (so called Frege/Hilbert formulations).

Equally we can set up e.g. a formal system for arithmetic which does everything with rules of inference (and uses a natural deduction logic again with no axioms).

We can't trade every rule of inference (all at once) for axioms ... or we'd never be able to infer anything. But we can trade axioms for rules of inference (boringly, we can trade the axiom $A$ for the rule from anything infer $A$).

So whatever distinction you want to make between (pure) FOL and a formal system/theory, a simple distinction between using axioms and using rules of inference isn't going to cut the cake in the right place.