How strong is Ramsey's theorem if infinity can be weak?

TL;DR

This paper investigates the logical strength of Ramsey's Theorem for various parameters within weak second-order arithmetic, revealing its equivalences, axiomatizations, and the impact of allowing $ riangle^0_2$-sets on its strength.

Contribution

It provides a detailed analysis of the first-order consequences of Ramsey's Theorem in weak arithmetic theories and introduces a $ riangle^0_2$-variant, expanding understanding of its logical strength.

Findings

RT$^n_k$ is equivalent to its relativization in certain models.

First-order consequences form a non-finitely axiomatizable subtheory of PA.

$ riangle^0_2$-RT$^2_2$ behaves similarly to RT$^2_2$ over certain base theories.

Abstract

We study the first-order consequences of Ramsey's Theorem for $k$-colourings of $n$-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm{RCA}^*_0$. Using the Chong-Mourad coding lemma, we show that in a model of $\mathrm{RCA}^*_0 + \neg \mathrm{I}\Sigma^0_1$, $\mathrm{RT}^n_k$ is equivalent to its own relativization to any proper $\Sigma^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe. We give an axiomatization of the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^n_k$ for $n \ge 3$. We show that they form a non-finitely axiomatizable subtheory of PA whose $\Pi_3$ fragment is $\mathrm{B}\Sigma_1 + \exp$ and whose $\Pi_{\ell+3}$ fragment for $\ell \ge 1$ lies between $\mathrm{I}\Sigma_\ell \Rightarrow \mathrm{B}\Sigma_{\ell+1}$ and $\mathrm{B}\Sigma_{\ell+1}$. We also consider the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^2_k$. We show that they form a subtheory of $\mathrm{I}\Sigma_2$ whose $\Pi_3$ fragment is $\mathrm{B}\Sigma_1 + \exp$ and whose $\Pi_4$ fragment is strictly weaker than $\mathrm{B}\Sigma_2$ but not contained in $\mathrm{I}\Sigma_1$. Additionally, we consider a principle $\Delta^0_2$-$\mathrm{RT}^2_2$, defined like $\mathrm{RT}^2_2$ but with both the $2$-colourings and the solutions allowed to be $\Delta^0_2$-sets. We show that the behaviour of $\Delta^0_2$-$\mathrm{RT}^2_2$ over $\mathrm{RCA}_0 + \mathrm{B}\Sigma^0_2$ is similar to that of $\mathrm{RT}^2_2$ over $\mathrm{RCA}^*_0$, and that $\mathrm{RCA}_0 + \mathrm{B}\Sigma^0_2 + \Delta^0_2$-$\mathrm{RT}^2_2$ is $\Pi_4$- but not $\Pi_5$-conservative over $\mathrm{B}\Sigma_2$. However, the statement we use to witness lack of $\Pi_5$-conservativity is not provable in $\mathrm{RCA}_0 +\mathrm{RT}^2_2$.