Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 76, Issue 1 (2011), 341-347.
A theorem on partial conservativity in arithmetic
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 The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it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.
J. Symbolic Logic, Volume 76, Issue 1 (2011), 341-347.
First available in Project Euclid: 4 January 2011
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Zentralblatt MATH identifier
Primary: 03F40: Gödel numberings and issues of incompleteness
Lindström, Per. A theorem on partial conservativity in arithmetic. J. Symbolic Logic 76 (2011), no. 1, 341--347. doi:10.2178/jsl/1294171003. https://projecteuclid.org/euclid.jsl/1294171003