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Flexible formulae and partial conservativity

Conference contribution
Authors Rasmus Blanck
Published in Logic Colloquium 2015, 3-8 August 2015, Helsinki, Finland
Pages 689
Publication year 2015
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 689
Language en
Links www.helsinki.fi/lc2015/fullprogram....
Keywords arithmetised metamathematics, flexible formulae, Pi_1-conservativity, end-extension
Subject categories Mathematical logic, Logic

Abstract

In [3], Woodin constructs an r.e. set Wₑ with the following feature: If M is any countable model of PA, and s is any M-finite set such that Wₑᴹ ⊆ s, then there is an end-extension N of M, such that N⊧PA, and Wₑᴺ = s. The set Wₑ has a distinct flavour of "flexibility" in the sense of e.g. Kripke [1] and Mostowski [2], who extend the first incompleteness theorem by constructing formulae whose "extensions as sets are left undetermined by the formal system". Moreover, Woodin implicitly establishes the Π₁-conservativity of T + Wₑ = s over T + Wₑ ⊆ s, which by the Orey-Hájek-Guaspari-Lindström characterisation allows the removal of the countability restriction from Woodin's theorem. In this talk, which reports on joint work with Ali Enayat, I give an overview of flexibility, and its relationship to Π₁-conservativity and interpretability. This includes some characterisations of Π₁-conservativity, and a discussion of how the relationship between these notions varies with the choice of base theory. [1] Saul A. Kripke, "Flexible" predicates of formal number theory, Proceedings of the American Mathematical Society, vol. 13 (1962), no. 4, pp. 647-650. [2] A. Mostowski, A generalization of the incompleteness theorem, Fundamenta Mathematicae, vol. 49 (1961), no. 2, pp. 205-232. [3] W. Hugh Woodin, A potential subtlety concerning the distinction between determinism and nondeterminism, Infinity: New Research Frontiers (Michael Heller and W. Hugh Woodin, editors), Cambridge University Press, 2011, pp. 119-129.

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