On Guaspari's problem about partially conservative sentences

TL;DR

This paper extends the understanding of partially conservative sentences over multiple theories, providing necessary and sufficient conditions for their existence and solving a specific case of Guaspari's problem.

Contribution

It generalizes Bennet's results to more than two theories and constructs sentences with specific unprovability and conservativity properties across finite families of theories.

Findings

Existence of $ ext{Pi}_n$ sentences unprovable in each theory

Existence of $ ext{Sigma}_n$ sentences conservative over each theory

Several non-implication results among properties of theory families

Abstract

We investigate sentences which are simultaneously partially conservative over several theories. First, we generalize Bennet's results on this topic to the case of more than two theories. In particular, for any finite family $\{T_i\}_{i \leq k}$ of consistent r.e. extensions of Peano Arithmetic, we give a necessary and sufficient condition for the existence of a $\Pi_n$ sentence which is unprovable in $T_i$ and $\Sigma_n$-conservative over $T_i$ for all $i \leq k$. Secondly, we prove that for any finite family of such theories, there exists a $\Sigma_n$ sentence which is simultaneously unprovable and $\Pi_n$-conservative over each of these theories. This constitutes a positive solution to a particular case of Guaspari's problem. Finally, we demonstrate several non-implications among related properties of families of theories.