An isomorphism theorem for models of Weak K\"onig's Lemma without primitive recursion

Abstract

We prove that if $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are countable models of the theory $\mathrm{WKL}^*_0$ such that $\mathrm{I}\Sigma_1(A)$ fails for some $A \in \mathcal{X} \cap \mathcal{Y}$, then $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are isomorphic. As a consequence, the analytic hierarchy collapses to $\Delta^1_1$ provably in $\mathrm{WKL}^*_0 + \neg\mathrm{I}\Sigma^0_1$, and $\mathrm{WKL}$ is the strongest $\Pi^1_2$ statement that is $\Pi^1_1$-conservative over $\mathrm{RCA}^*_0 + \neg\mathrm{I}\Sigma^0_1$. Applying our results to the $\Delta^0_n$-definable sets in models of $\mathrm{RCA}^*_0 + \mathrm{B}\Sigma^0_n + \neg\mathrm{I}\Sigma^0_n$ that also satisfy an appropriate relativization of Weak K\"onig's Lemma, we prove that for each $n \ge 1$, the set of $\Pi^1_2$ sentences that are $\Pi^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}\Sigma^0_n + \neg\mathrm{I}\Sigma^0_n$ is c.e. In contrast, we prove that the set of $\Pi^1_2$ sentences that are $\Pi^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}\Sigma^0_n$ is $\Pi_2$-complete. This answers a question of Towsner. We also show that $\mathrm{RCA}_0 + \mathrm{RT}^2_2$ is $\Pi^1_1$-conservative over $\mathrm{B}\Sigma^0_2$ if and only if it is conservative over $\mathrm{B}\Sigma^0_2$ with respect to $\forall \Pi^0_5$ sentences.