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

TL;DR

This paper establishes an isomorphism theorem for models of Weak K"onig's Lemma without primitive recursion, revealing collapse results in the analytic hierarchy and analyzing the conservativity of certain subsystems over base theories.

Contribution

It proves an isomorphism theorem for countable models of WKL*0 under specific failure conditions of IΣ₁, and characterizes the complexity of conservative extensions over various subsystems.

Findings

Analytic hierarchy collapses to Δ¹₁ in certain models.

WKL is the strongest Π¹₂ statement that is Π¹₁-conservative over RCA*₀ + ¬IΣ₁.

The set of Π¹₂ sentences that are Π¹₁-conservative over certain theories is c.e. or Π₂-complete.

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.