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MARGINALIA ON A THEOREM OF WOODIN

Published online by Cambridge University Press:  21 March 2017

RASMUS BLANCK
Affiliation:
DEPARTMENT OF PHILOSOPHY, LINGUISTICS, AND THEORY OF SCIENCE UNIVERSITY OF GOTHENBURG BOX 200, SE 405 30, GOTHENBURG, SWEDEN E-mail: rasmus.blanck@gu.se
ALI ENAYAT
Affiliation:
DEPARTMENT OF PHILOSOPHY, LINGUISTICS, AND THEORY OF SCIENCE UNIVERSITY OF GOTHENBURG BOX 200, SE 405 30, GOTHENBURG, SWEDEN E-mail: ali.enayat@gu.se
Corresponding

Abstract

Let $\left\langle {{W_n}:n \in \omega } \right\rangle$ 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 $e \in \omega$ (that depends on T) with the property that if ${\cal M}$ is a countable model of T and for some ${\cal M}$ -finite set s, ${\cal M}$ satisfies ${W_e} \subseteq s$ , then ${\cal M}$ has an end extension ${\cal N}$ that satisfies T + W e = s.

Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment ${{\rm{I}\rm{\Sigma }}_1}$ of PA, and remove the countability restriction on ${\cal M}$ 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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Copyright © The Association for Symbolic Logic 2017 

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