A variety of partially conservative sentences

TL;DR

This paper investigates the existence of sentences that are conservative over different extensions of Peano Arithmetic, providing affirmative answers to longstanding questions about their properties and existence.

Contribution

It proves the existence of sentences that are hereditarily conservative over any single theory, answering Guaspari's question affirmatively.

Findings

Existence of $ heta$ sentences that are $ heta$-conservative over various theories

Existence of sentences that are hereditarily $ heta$-conservative over any single theory

Affirmative resolution of Guaspari's question about conservative sentences

Abstract

We study the existence of a $\Theta$ sentence which is simultaneously $\Gamma$-conservative over consistent RE extensions $T$ and $U$ of Peano Arithmetic for various reasonable pairs $(\Gamma, \Theta)$. As a result of this study, we prove the existence of a sentence which is essentially $\Theta$ and exactly hereditarily $\Gamma$-conservative over any single theory $T$ for various reasonable pairs $(\Gamma, \Theta)$. This is an affirmative answer to Guaspari's question.