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 < q. 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 The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it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