2021/12/20 by Fiori-Carones, Marta, Kołodziejczyk, Leszek Aleksander, Wong, Tin Lok +1 · 4 citations
#03B30 #03C10 #03C62 #03F30 #03F35 #03H15 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2112.10876
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 + \negIΣ01, and WKL is the strongest Π12 statement that is Π11-conservative over RCA^*0 + \negIΣ01. Applying our results to the Δ0n-definable sets in models of RCA^*0 + BΣ0n + \negIΣ0n 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 + \negIΣ0n is c.e. 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.