Really confused about the relationship between set theory, functions, ZFC, Peano axioms, etc.

Solution 1:

Peano's name is attached to two different theories about the natural numbers, which unfortunately don't always have clearly different names. The following convention is fairly common, though:

• The "Peano axioms" is a second-order theory, which just describes the successor function and a general induction axiom. With some amount of set theory as a background feature of the logic, we can then define addition and multiplication without needing specific axioms for them.

• "Peano Arithmetic" is a first-order theory, developed long after Peano's time as a "best-effort" first-order approximation of the second-order Peano axioms. It has specific axioms for the successor function and addition and multiplication, and an induction axiom schema that only works for properties that can be expressed in its first order language of successor+addition+multiplication.

Peano Arithmetic is what is usually meant by just the abbreviation PA. (Note capital A and no "the" for PA).

Because the induction axiom in Peano Arithmetic is not as strong as the full second-order induction axiom, the theory is weaker -- it has models that are not isomorphic to the usual $\mathbb N$. (It is hard-to-impossible to describe one of these non-standard models; we just have an existence proof for them. It depends crucially on the fact that first-order logic is complete: every consistent theory has a model. This is not true about the standard semantics for second-order logic, which is why the second-order axioms are stronger).

Despite being weaker, first-order PA has a lot more theoretical interest, because first-order logic is a lot better behaved than second-order.

For "don't functions require ZFC?", see When does the set enter set theory? or perhaps What is the dependency hierarchy in foundational mathematics?.