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Flexibility and Π₁-conservativity

Conference contribution
Authors Rasmus Blanck
Published in Bristol-Gothenburg-Oxford Logic and Set Theory Postgraduate Student Day, 22 April 2015, Bristol.
Publication year 2015
Published at Department of Philosophy, Linguistics and Theory of Science
Language en
Keywords arithmetised metamathematics, flexible formulae, Pi_1-conservativity, model theory
Subject categories Logic, Mathematical logic

Abstract

In a somewhat recent paper (2011), Woodin constructs an r.e. set W with the following feature: If M is any countable model of PA, S is any finite set, and W (as interpreted in M) is a subset of S, then there is an end-extension N of M, such that N is a model of PA, and W (as interpreted in N) is equal to S. The set W has a distinct flavour of "flexibility" in the sense of e.g. Mostowski and Kripke, who in the early 1960s extended the first incompleteness theorem by constructing formulae whose "extensions as sets are left undetermined by the formal system". In general, these flexibility results are verifiable in rather weak subsystems of arithmetic. On the other hand, Woodin's construction also establishes the Pi_1-conservativity of PA +"W equals S" over PA +"W is a subset of S", which by the Orey-Hájek-Guaspari-Lindström characterisation provides an interpretation of the former theory in the latter. In this talk, which reports on joint work with Ali Enayat, I will give an overview of flexibility, and its relationship to Pi_1-conservativity and interpretability. This includes some characterisations of Pi_1-conservativity. I will also discuss how the relationship between these notions varies with the choice of base theory.

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