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Regarding the small Veblen ordinal, Rathjen and Weiermann gave an analysis of theories in that range of strength in Proof-theoretic investigations of Kruskal's theorem.1 Working over a reasonable bas …

Yes: for many theories, $\alpha$ is the proof-theoretic ordinal of T exactly when T proves that $f_\beta$ is total for all $\beta<\alpha$, but does not prove $f_\alpha$ is total. (Where $f_\alpha$ is …

To state the general rule: when counting quantifier changes, you ignore bounded quantifiers that come after all unbounded ones. But even a bounded quantifier, if it comes before an unbounded one, m …

There's a good reason that concrete examples are going to be rare. Existence of a first-order proof (at least in a countable language) is a $\Sigma_1$ property in the language of arithmetic, and it's …

I think usually one adds the condition that $\phi$ be a $\Delta_1$ (i.e. computable) formula. As Gro-Tsen has pointed out, the answer is no: there are lots of functions which are provably total, domi …

I want to bound some functions using the fast-growing hierarchy, but for accounting reasons it looks like it's going to be easier to deal with a modified hierarchy that grows at "$1/\omega$-th" the ra …

If $\alpha$ is a reasonable presentation of a computable ordinal then the proof-theoretic ordinal of $ACA_0+\alpha$ is well-ordered is the smallest $\epsilon$ number $>\alpha$. (There's a proof of th …

First, by "stronger than", Girard means "at least as strong as". So the identity rule can be read as saying "If you have a C on the left side, you can have C on the right side as well (because you ca …

The witness coding a proof of 0=1 in a nonstandard model is likely to be very specific; depending on your encoding, most nonstandard numbers may not code proofs at all. And even if all numbers encode …

I suspect the paper you want is Avigad and Sommer, A Model-Theoretic Approach to Ordinal Analysis. As the name suggests, they give an ordinal analysis rooted in the structure of models of arithmetic. …

I'm not an expert (I've played around with HOL a few times, but never fully formalized an interesting theorem), but my experience was that it's similar to writing up a technical proof after you've wor …

The proof-theoretic ordinal of any theory is less than $\omega_1^{CK}$. No notations are known for second-order arithmetic, let alone ZFC.

I think the short answer is no, and I'm not sure there really could be: I don't think we "know" any ordinals above the ordinal of $\Pi^1_2-CA$ but below $\omega_1^{CK}$. Theoretically someone could w …

As Noah says, the direct successor of Gentzen's method, cut-elimination, has been generalized up to $\Pi^1_2$-comprehension. This was shown separately by Rathjen and Arai; the full results have never …

This is an instance of a general phenomenon: adding true $\Pi_1$ sentences to a reasonable theory doesn't change its proof-theoretic ordinal. $Con(T)$ is $\Pi_1$, so if $T$ is any consistent theory, …