Definability over \mathrm BΣ⁰₂-models

2025/10/21 by Chong, Chi Tat, Wong, Tin Lok
#03C62 (Secondary) #03D55 (Primary) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2510.18490

Abstract

Let \mathfrak M=(M,\mathcal X) be a model of RCA₀+Σ⁰₂-bounding in which Σ⁰₂(A)-induction fails for some A∈\mathcal X. We show that (i) if \mathfrak M is a model of the combinatorial principle Ramsey's Theorem for Pairs, the Cohesive Set Theorem or the Tree Theorem, then there is a Δ⁰₁(A)-instance of the principle with no solution in \mathfrak M that is arithmetically definable relative to A; and (ii) any set of minimal Turing degree in \mathfrak M that is arithmetically definable relative to A has Turing jump equivalent to A'.

Citations

• An isomorphism theorem for models of Weak König's Lemma without primitive recursion
• The proof-theoretic strength of Ramsey's theorem for pairs and two colors

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• Cohesive avoidance and arithmetical sets
• Weaker cousins of Ramsey's theorem over a weak base theory
• Degrees bounding principles and universal instances in reverse\n mathematics
• Bi-Isolated d.c.e. Degrees and Σ₁ Induction
• Combinatorial principles equivalent to weak induction