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 IΣ₁(A) fails for some A ∈ X∩ Y , then (M,X) and (M,Y) are isomorphic. As a consequence, the analytic hierarchy collapses to Δ¹₁ provably in WKL*₀ + IΣ⁰₁ , and WKL is the strongest Π¹₂ statement that is Π¹₁ -conservative over RCA*₀ + IΣ⁰₁ . Applying our results to the Δ⁰ₙ -definable sets in models of RCA*₀ + BΣ⁰ₙ + IΣ⁰ₙ that also satisfy an appropriate relativization of weak König’s lemma, we prove that for each n ≥ 1 , the set of Π¹₂ sentences that are Π¹₁ -conservative over RCA*₀ + BΣ⁰ₙ + IΣ⁰ₙ is computably enumerable. In contrast, we prove that the set of Π¹₂ sentences that are Π¹₁ -conservative over RCA*₀ + BΣ⁰ₙ is Π₂ -complete. This answers a question of Towsner. We also show that RCA₀ + RT²₂ is Π¹₁ -conservative over BΣ⁰₂ if and only if it is conservative over BΣ⁰₂ with respect to ∀ Π⁰₅ sentences.