A variety of partially conservative sentences
Haruka Kogure, Taishi Kurahashi
TL;DR
This work extends the theory of partial conservativity by introducing and exploiting the novel notion of Σ_n ↓ Π_n-conservativity, situating it within the landscape of hereditary and exact conservativity across pairs of theories. It generalizes and unifies prior results (e.g., Bennet, KOSV, Guaspari) by analyzing a wide spectrum of Conservativity notions (Σ_n, Π_n, Δ_n, Σ_n ∧ Π_n, and Boolean closures) and their simultaneous behavior over two theories. The authors prove that for any RE consistent extension T of PA and any reasonable pair (Γ, Θ), there exists a sentence that is essentially Θ and exactly hereditarily Γ-conservative over T, providing an affirmative answer to Guaspari’s second question in a broad setting. They also develop a comprehensive taxonomy of theory-pair conditions (B_n, C_n, C^−_n, C^B_n, etc.) governing when such sentences exist, and they supply constructive arguments via fixed-point techniques and provability predicates to realize these sentences. The results illuminate a wide class of independent sentences with finely tuned conservativity properties, highlighting a rich interplay between proof-theoretic strength and model-theoretic conservativity in arithmetic.)
Abstract
We study the existence of a $Θ$ sentence which is simultaneously $Γ$-conservative over consistent RE extensions $T$ and $U$ of Peano Arithmetic for various reasonable pairs $(Γ, Θ)$. As a result of this study, we prove the existence of a sentence which is essentially $Θ$ and exactly hereditarily $Γ$-conservative over any single theory $T$ for various reasonable pairs $(Γ, Θ)$. This is an affirmative answer to Guaspari's question.