Let $\mathbb{N}$ be the standard model of the natural numbers. For any statement in the language of arithmetic, we can translate into a statement in the language of set theory by asking if it is true of $\mathbb{N}$.

Let's say that a statement in arithmetic is "extraneous" if it is independent of PA. For example, ZFC proves Con(PA), which is extraneous.

My question is, is there a set of statements $S$ (in the language of set theory), such that $S$ proves no extraneous statements, and $S+PA=ZFC$ (or perhaps $S+PA \vdash ZFC$).

Edit: We can also consider the same question, but with PA replaced with the set of arithmetical statements provable in ZFC.