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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 …

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 …

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

As Noah pointed out, if it's provable, there must be an algorithm for cut-elimination for STT because the underlying statement is $\Pi_2$ (for any deduction in STT, there is a corresponding deduction …

To add to Joel's answer, in the most common theories (for instance, ordinary first-order logic), these are equivalent (by the deduction theorem), but there are plenty of theories where they aren't. O …

I'm wondering if a particular theory of second order arithmetic has been studied or is known to be equivalent to some other theory. Consider the formulas generated by $\Pi^1_1$ and $\Sigma^1_1$ formu …

I hope it's not too tacky to answer my own question now that I've had a few days and train rides to think about it. The answer is that the proof theoretic ordinal of $\Delta-TI_0$ is the Howard-Bachm …

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 …

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 …

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 think you'll have to at least tweak the definition of your hierarchy, since a (strong enough) consistent theory can't prove that it doesn't prove something: $T$ can prove "$\neg Con(T)$ implies that …

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 …

Historically, reverse math is very closely tied to ideas in proof theory, but as Andreas points out, over the last decade or so, the connection to recursion theory has been very strong. For the found …

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, …

I believe BISH includes the Fan Theorem, which, being the contrapositive of weak König's lemma, is not provable in $RCA_0$.