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}\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 König’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 computably enumerable. 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.