According to this Wiki article, ZFC-R (ZFC without the Axioms of Replacement) cannot construct $\omega \cdot 2$. It also claims that $V_{\omega\cdot 2}$ is a model of ZFC-R and is sufficient to do basically all of second-order arithmetic.
(It also seems to refer to ZFC-R as Z, which isn't clear to me that they're equivalent.)
However, in this article it says that the proof-theoretic ordinal of a theory is the smallest recursive ordinal that the theory can't prove to be well-founded. It then further claims that the proof-theoretical ordinal of second-order arithmetic is (currently) undescribably large--far larger than $\omega \cdot 2$. Since ZFC-R can do second-order arithmetic, its proof-theoretical ordinal must be at least as large.
So how is it that ZFC-R can prove that $\omega\cdot 2$ is well-founded, but apparently cannot prove it exists?