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The proof-theoretic strength of a theory is measured by the $\mathsf{\Pi}_{1}^{1}$-ordinal of the theory and it is called the proof-theoretic ordinal (PTO) of the theory (there are other ordinal analyses, like the $\Pi_{2}^{0}$-ordinal of the theory).

My understanding of the bounding scheme is rudimentary. But it is an induction principle after all, just for a restricted class of formulas. Hence, it appears to me that the PTO of $\mathsf{B\Sigma}_{2}^{0}$ (equivalently, $\mathsf{B\Pi}_{1}^{0}$) should be at least that of $\mathsf{RCA}_{0}$, i.e. $\omega^{\omega}$. On the other hand, it should not be larger than $\varepsilon_{0}$ since two facts are known: First, for each exponent $p\ge 3$, the infinite Ramsey's Theorem for exponent $p$ (denoted by $\mathrm{IRT}(p)$) is equivalent to $\mathsf{ACA}_{0}$ over $\mathsf{RCA}_{0}$. Second, over $\mathsf{RCA}_{0}$, $\mathrm{IRT}(1)$ and $\mathsf{B\Sigma}_{2}^{0}$ are equivalent. But what is precisely the PTO of $\mathsf{B\Sigma}_{2}^{0}$?

In order to be clear, by $\mathrm{IRP}(p)$ I mean the statement that for all $r\in\mathbb{N}$ and all functions $c:[\mathbb{N}]^{p}\rightarrow [r]$, there exists and infinite subset $H\subseteq\mathbb{N}$ such that $c|_{[H]^{p}}$ is constant. Here, $[r]$ denotes the set $\{1,\ldots,r\}$ and $[\mathbb{N}]^{p}$ denotes the set of all subsets of $\mathbb{N}$ of size $p$.

John
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  • On Kaye's book Corollary 10.9 you have that $\textbf{B}\Sigma^0_{n+1}$ is $\Pi^0_{n+2}$ conservative over $\textbf{I}\Sigma^0_n$. I assume this would imply that the proof theoretic ordinal of $\textbf{I}\Sigma^0_1$ should be the same as $\textbf{B}\Sigma^0_{2}$. – Giorgio Genovesi Sep 08 '24 at 15:43

1 Answers1

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I originally added the following as an update to the question. I decided to post it as an answer since no objections have been raised so far. That way the question will be removed from the unanswered list. In any cae, here is my attempt at answering the question.

Over $\mathsf{PA}^{-}+\mathsf{I\Sigma}_{0}^{0}$ (see Theorem A in 1) we have the following non-reversible implications: $$\mathsf{I\Sigma}_{2}^{0}\Rightarrow\mathsf{B\Sigma}_{2}^{0}\Rightarrow\mathsf{I\Sigma}_{1}^{0}$$

Now, the PTO of $\mathsf{I\Sigma}_{2}^{0}$ is $\omega^{\omega^{\omega}}$, while the PTO of $\mathsf{I\Sigma}_{1}^{0}$ is $\omega^{\omega}$. Therefore, $\mathsf{B\Sigma}_{2}^{0}$ has PTO ordinal $\alpha$ with $\omega^{\omega}\leq\alpha\leq\omega^{\omega^{\omega}}$. In fact, in accordance with this answer (note $\mathsf{\Pi}_{2}^{0}$ ordinal analysis is used there) $\alpha$ may very well be either $\omega^{\omega}$ or $\omega^{\omega^{\omega}}$.

  1. Paris, J. B.; Kirby, L. A. S., $\mathsf{\Sigma}_{n}$-collection schemas in arithmetic, Logic colloquium ’77, Proc., Wroclaw 1977, Stud. Logic Found. Math. Vol. 96, 199-209 (1978). ZBL0442.03042.
John
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