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Marginalia on a theorem of Woodin

Journal article
Authors Rasmus Blanck
Ali Enayat
Published in Journal of Symbolic Logic
Volume 82
Issue 1
Pages 359-374
ISSN 0022-4812
Publication year 2017
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 359-374
Language en
Links https://doi.org/10.1017/jsl.2016.8
Keywords Turing machines, Peano arithmetic, nonstandard models, flexible predicates, Kolmogorov complexity.
Subject categories Logic, Mathematical logic

Abstract

Let ⟨Wn : n ∈ ω⟩ be a canonical enumeration of recursively enumerable sets, and suppose T is a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index e∈ω (that depends on T) with the property that if M is a countable model of T and for some M-finite set s, M satisfies We⊆s, then M has an end extension N that satisfies T + We=s. Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment IΣ1 of PA, and remove the countability restriction on M when T extends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.

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