Can we use metatheory of set theory for proof statements in set theory?

Question

I wanna know if is possible or have examples of theorems in set theory, for example $\beta$, that have a demonstration of forme $Cons(ZFC)\Rightarrow \beta$ but $\beta$ is independent of axioms of $ZFC$.

Answer 1

Yes, it is possible. You can add $Con(ZFC)$ as an assumption. If you then derive $\beta$ you have proven within $ZFC$ that $Con(ZFC)\implies \beta$. If you can then prove that $\beta$ is not provable from $ZFC$ you are there.

A trivial example is to take $\beta = Con(ZFC)$. Then ordinary logic proves $Con(ZFC) \implies \beta$ even without $ZFC$.