Why in ZFC any expressible set also exists?

Question

I would like to ask you a few questions about the following quote from Wang (On Denumerable Bases, 1955):

"Zermelo's set theory is, for example, a system in which every number set, if expressible in the system, can be proved to exist in the system."

1) What is a number set?

2) Can someone give me more information on the link between expressibility and existence in ZFC?

3) How the proof mentioned by Wang works?

Thank you!

Answer 1

See Thoralf Skolem et alii (editors), Mathematical Interpretation of Formal Systems (North Holland, 1955), page 61 :

number set (i.e., set of positive integers).

In that context, "$S$ is expressible" means : $∀m(m∈S ↔ \varphi(m))$ for some formula $\varphi$.

Wang (page 59) refers to Bernays-Godel set theory, Quine-Hailperin theory (ordinarily known as the "New Foundations") as well as Zermelo's set theory and Zermelo-Fraenkel set theory (page 64).

Zermelo-Fraenkel Set Theory, using the Infinity Axiom, proves the existence of the set $\omega$ of natural numebrs.

Thus, it is enough to apply the Separation Schema to $\omega$ to prove the existence of $S$ :

$\exists S \ \forall m \ (m \in S \leftrightarrow m \in \omega \land \varphi(m))$.