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  • RASMUS BLANCK (a1) and ALI ENAYAT (a2)

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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L. Beklemishev and A. Visser , On the limit existence principles in elementary arithmetic and ${\rm{\Sigma }}_n^0$ -consequences of theories . Annals of Pure and Applied Logic, vol. 136 (2005), pp. 5674.

C. Dimitracopoulos and J. Paris , A note on a theorem of H. Friedman . Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), pp. 1317.

D. Guaspari , Partially conservative extensions of arithmetic . Transaction of the American Mathematical Society, vol. 254 (1979), pp. 4768.

P. Hájek and P. Pudlák , Metamathematics of First-Order Arithmetic, Springer-Verlag, Berlin, 1993.

R. Kossak and J. Schmerl , The Structure of Models of Peano Arithmetic, Oxford University Press, Oxford, 2006.

P. Lindström , Aspects of Incompleteness, Lecture Notes in Logic, vol. 10, ASL Publications, 1997.

J.-P. Ressayre , Nonstandard universes with strong embeddings, and their finite approximations , Logic and Combinatorics, Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1987, pp. 333358.

S. Simpson , Subsystems of Second Order Arithmetic, Springer-Verlag, Berlin, 1999.

C. Smoryński , Modal Logic and Self-reference, Springer-Verlag, Berlin, 1985.

W. H. Woodin , A potential subtlety concerning the distinction between determinism and nondeterminism , Infinity, New Research Frontiers ( M. Heller and W. H. Woodin , editors), Cambridge University Press, Cambridge, 2011, pp. 119129.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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