Improving on a result of Arana, we construct an effective family (φr| r∈ℚ∩[0,1]) of Σn-conservative Πn sentences, increasing in strength as r decreases, with the property that ¬φp is Πn-conservative over PA+φq whenever p <. We also construct a family of Σn sentences with properties as above except that the roles of Σn and Πn 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 The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it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.
Per Lindström. "A theorem on partial conservativity in arithmetic." J. Symbolic Logic 76 (1) 341 - 347, March 2011. https://doi.org/10.2178/jsl/1294171003