A theorem on partial conservativity in arithmetic

Abstract

Improving on a result of Arana, we construct an effective family (φ_r| r∈ℚ∩[0,1]) of Σ_n-conservative Π_n sentences, increasing in strength as r decreases, with the property that ¬φ_p is Π_n-conservative over PA+φ_q whenever p <. We also construct a family of Σ_n sentences with properties as above except that the roles of Σ_n and Π_n are reversed. The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it to analogues of the E-tree at higher levels of the arithmetical hierarchy.