Doubly partially conservative sentences

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 $\hat{I}˜$ sentences that are doubly $(\hat{I}“,\hat{I}›)$-conservative over $T$ for several triples $(\hat{I}˜,\hat{I}“,\hat{I}›)$. Among other things, we prove that the existence of a $\hat{I}{”}_{n+1}(\mathsf{PA})$ sentence that is doubly $(\hat{I}{\pounds }_n,\hat{I}{\pounds }_n)$-conservative over $T$ is equivalent to the $\hat{I}{\pounds }_{n+1}$-inconsistency of $T$ over $\mathsf{PA}$. Secondly, we study $\hat{I}˜$ sentences that are hereditarily doubly $(\hat{I}“,\hat{I}›)$-conservative over $T$ for several triples $(\hat{I}˜,\hat{I}“,\hat{I}›)$.

25 pages