Iterative forcing and hyperimmunity in reverse mathematics

Ludovic Patey 1, 2
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Tows-ner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperim-munity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions. More than the actual separations, we provide a systematic method to design a computability-theoretic property which enables one to distinguish two statements, based on an analysis of their combinatorics.
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Ludovic Patey. Iterative forcing and hyperimmunity in reverse mathematics. Computability, IOS Press, 2017, 6 (3), pp.209 - 221. ⟨10.3233/COM-160062⟩. ⟨hal-01888601⟩

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