Doubly partially conservative sentences
Haruka Kogure, Taishi Kurahashi
TL;DR
This work advances the understanding of Solovay-type results by systematically examining doubly conservative sentences across multiple arithmetical levels and their hereditary versions. It introduces and leverages relativized conservativity notions, $\Delta_{n+1}(U)$-sentences, and witness-comparison predicates to establish precise equivalences between the existence of such sentences and relativized consistency conditions like $\Sigma_{n+1}$-consistency over $PA$. The authors provide a network of results, including a central equivalence: a $\Delta_{n+1}(PA)$ sentence that is doubly $(\Sigma_n,\Sigma_n)$-conservative over $T$ exists exactly when $T$ is not $\Sigma_{n+1}$-consistent over $PA$, and they extend this analysis to several triples $(\Theta,\Gamma,\Lambda)$ with comprehensive tables. They further develop hereditary variants, linking the existence of horodical doubly conservative sentences to $\Sigma_n{\downarrow}\Pi_n$-conservativity, and they present both constructive and nonexistence results (notably for $\Delta_{n+1}(PA)$). Altogether, the paper maps the boundary between provability-conservativity and relative consistency, clarifying when strong double-conservativity manifestations can occur and how they propagate through subtheories of arithmetic.
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 $Θ$ sentences that are doubly $(Γ, Λ)$-conservative over $T$ for several triples $(Θ, Γ, Λ)$. Among other things, we prove that the existence of a $Δ_{n+1}(\mathsf{PA})$ sentence that is doubly $(Σ_n, Σ_n)$-conservative over $T$ is equivalent to the $Σ_{n+1}$-inconsistency of $T$ over $\mathsf{PA}$. Secondly, we study $Θ$ sentences that are hereditarily doubly $(Γ, Λ)$-conservative over $T$ for several triples $(Θ, Γ, Λ)$.