How Strong Is Ramsey’s Theorem if Infinity Can Be Weak?

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 $\mathrm {RCA}^*_0$ that does not satisfy $\Sigma ^0_1$ induction, $\mathrm {RT}^n_k$ is equivalent to its 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 a complete 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 $\mathrm {PA}$ whose $\Pi _3$ fragment coincides with $\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 give a complete axiomatization of the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$ . In general, we show that the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$ form a subtheory of $\mathrm {I} \Sigma _2$ whose $\Pi _3$ fragment coincides with $\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 $\Delta ^0_2$ - $\mathrm {RT}^2_2$ which is defined like $\mathrm {RT}^2_2$ but with both the $2$ -colourings and the solutions allowed to be $\Delta ^0_2$ -sets rather than just sets. We show that the behaviour of $\Delta ^0_2$ - $\mathrm {RT}^2_2$ over $\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$ is in many ways analogous 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 failure of $\Pi _5$ -conservativity is not provable in $\mathrm {RCA}_0 +\mathrm {RT}^2_2$ .