An isomorphism theorem for models of Weak König's Lemma without primitive recursion

We prove that if $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are countable models of the theory $\mathrm{WKL}^*_0$ such that $\mathrm{I}Σ_1(A)$ 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 $Δ^1_1$ provably in $\mathrm{WKL}^*_0 + \neg\mathrm{I}Σ^0_1$, and $\mathrm{WKL}$ is the strongest $Π^1_2$ statement that is $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \neg\mathrm{I}Σ^0_1$. Applying our results to the $Δ^0_n$-definable sets in models of $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n + \neg\mathrm{I}Σ^0_n$ that also satisfy an appropriate relativization of Weak König's Lemma, we prove that for each $n \ge 1$, the set of $Π^1_2$ sentences that are $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n + \neg\mathrm{I}Σ^0_n$ is c.e. In contrast, we prove that the set of $Π^1_2$ sentences that are $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n$ is $Π_2$-complete. This answers a question of Towsner. We also show that $\mathrm{RCA}_0 + \mathrm{RT}^2_2$ is $Π^1_1$-conservative over $\mathrm{B}Σ^0_2$ if and only if it is conservative over $\mathrm{B}Σ^0_2$ with respect to $\forall Π^0_5$ sentences.

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