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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?

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You need to distinguish between well ordered sets and von Neumann ordinals. While $\sf ZFC-R$ cannot prove the existence of von Neumann ordinals $\geq\omega+\omega$, it can prove the existence of many well ordered sets. In fact, in $V_{\omega+\omega}$ any order type below $\beth_\omega$.

The term proof theoretic ordinal has a technical meaning to it, and it is in fact a countable ordinal. It is not the term for the largest von Neumann ordinal that a theory can prove to exist, though.

(To wit, if $M$ is any countable transitive model of $\sf ZFC-R$, then it contains only countably many ordinals, and countably many sets. So only countably many order types exist there. And we could ask what is the least order type of a set that provably exists in a transitive model of $\sf ZFC-R$. This has to be a countable ordinal.)

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    +1. For the OP: elaborating on the first sentence of this answer's second paragraph, the proof-theoretic ordinal is defined in terms of *computable linear orders* which the theory in question can prove are well-founded; this is a very different thing from *definable ordinals* which the theory can prove exist. There are lots of ways to ask, "How large are the ordinals that this theory can 'get'?", and they're not equivalent. For yet a third, look at [this paper by Arai](https://arxiv.org/abs/1101.5660). – 2017-01-11
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    @NoahSchweber So, if I understand you correctly, ZFC-R can prove that sets with the order type $\omega\cdot 2$ exist, but not that the specific set $\{\omega + n: n\in N\}$ exists? – 2017-01-11
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    @eyeballfrog Exactly. For instance, the set $$\{(m, n): \mbox{$m$ is even and $n$ is odd, or $m$ and $n$ have the same parity and $m – 2017-01-11
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    @eyeballfrog: Yes. – 2017-01-11
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    And I guess one final question, which might seem obvious given the above, but I want to be sure I understand. There exists some ordinal (though we don't know what it is yet) that ZFC-R cannot prove any sets with that order type exist, right? – 2017-01-11
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    @eyeballfrog: Clearly $\beth_\omega$ is an example, since no set in $V_{\omega+\omega}$ has cardinality $\beth_\omega$. Notice, however, that this is also a difference between internal and external definitions. It might be that $M$ is a countable model of ZFC-R, but then it only has countably many sets with countably many order types. But internally, ZFC-R can still prove the existence of the real numbers, their well-ordering, and any finite iteration of taking power sets. So "Internal $\beth_\omega$" seems to be a reasonable limit, but which ordinal is that? We can't quite say. – 2017-01-12