An isomorphism theorem for models of weak König’s lemma without primitive recursion

We prove that if (M, X) and (M, Y) are countable models of the theory WKL^*₀ such that I₁ (A) fails for some A X Y, then (M, X) and (M, Y) are isomorphic. As a consequence, the analytic hierarchy collapses to ^1₁ provably in WKL^*₀ + I^0₁, and WKL is the strongest ^1₂ statement that is ^1₁ -conservative over RCA^*₀ + I^0₁. Applying our results to the ^0₍ -definable sets in models of RCA^*₀ + B^0₍ + I^0₍ that also satisfy an appropriate relativization of weak König’s lemma, we prove that for each n 1, the set of ^1₂ sentences that are ^1₁ -conservative over RCA^*₀ + B^0₍ + I^0₍ is computably enumerable. In contrast, we prove that the set of ^1₂ sentences that are ^1₁ -conservative over RCA^*₀ + B^0₍ is ₂ -complete. This answers a question of Towsner. We also show that RCA₀ + RT^2₂ is ^1₁ -conservative over B^0₂ if and only if it is conservative over B^0₂ with respect to ^0₅ sentences.