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arXiv · 2503.12373

Doubly partially conservative sentences

Haruka Kogure·Taishi Kurahashi
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Abstract

The purpose of the present paper is to analyze several variants of Solovay's theorem on the existence of doubly partially conservative sentences. First, we investigate $\Theta$ sentences that are doubly $(\Gamma, \Lambda)$-conservative over $T$ for several triples $(\Theta, \Gamma, \Lambda)$. Among other things, we prove that the existence of a $\Delta_{n+1}(\mathsf{PA})$ sentence that is doubly $(\Sigma_n, \Sigma_n)$-conservative over $T$ is equivalent to the $\Sigma_{n+1}$-inconsistency of $T$ over $\mathsf{PA}$. Secondly, we study $\Theta$ sentences that are hereditarily doubly $(\Gamma, \Lambda)$-conservative over $T$ for several triples $(\Theta, \Gamma, \Lambda)$.

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Haruka Kogure, Taishi Kurahashi. 2025-03-16. Doubly partially conservative sentences. https://arxiv.org/abs/2503.12373

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