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MARGINALIA ON A THEOREM OF WOODIN

  • RASMUS BLANCK (a1) and ALI ENAYAT (a2)
Abstract
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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[1] AvigadJ., Formalizing forcing arguments in subsystems of second-order arithmetic . Annals of Pure and Applied Logic, vol. 82 (1996), pp. 165191.
[2] BeklemishevL. and VisserA., 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.
[3] BlanckR., Two consequences of Kripke’s lemma , Idées Fixes (KasåMartin, editor), University of Gothenburg Publications, 2014, pp. 4553.
[4] BoolosG., The Logic of Provability, Cambridge University Press, Cambridge, 1979.
[5] CornarosC., Versions of Friedman’s theorem for fragments of PA, http://www.softlab.ntua.gr/∼nickie/tmp/pls5/cornaros.pdf.
[6] D’AquinoP., A sharpened version of McAloon’s theorem on initial segments of0 . Annals of Pure and Applied Logic, vol. 61 (1993), pp. 4962.
[7] DimitracopoulosC. and ParisJ., A note on a theorem of H. Friedman . Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), pp. 1317.
[8] DimitracopoulosC. and PaschalisV., End extensions of models of weak arithmetic theories . Notre Dame Journal of Formal Logic, to appear.
[9] EnayatA. and WongT., Model theory of ${\rm{WKL}}_0^{^{\rm{*}}}$ . Annals of Pure and Applied Logic, to appear.
[10] GuaspariD., Partially conservative extensions of arithmetic . Transaction of the American Mathematical Society, vol. 254 (1979), pp. 4768.
[11] HájekP., Interpretability and fragments of arithmetic . Arithmetic, Proof Theory, and Computational Complexity, Oxford University Press, Oxford, 1993, pp. 185196.
[12] HájekP. and PudlákP., Metamathematics of First-Order Arithmetic, Springer-Verlag, Berlin, 1993.
[13] JaparidzeG. and de JonghD., The logic of provability , Handbook of Proof Theory, North-Holland, Amsterdam, 1998, pp. 475546.
[14] KayeR., Models of Peano Arithmetic, Oxford University Press, Oxford, 1991.
[15] KossakR. and SchmerlJ., The Structure of Models of Peano Arithmetic, Oxford University Press, Oxford, 2006.
[16] KripkeS., “Flexible” predicates of formal number theory . Proceedings of the American Mathematical Society, vol. 13 (1962), pp. 647650.
[17] LindströmP., Aspects of Incompleteness, Lecture Notes in Logic, vol. 10, ASL Publications, 1997.
[18] McAloonK., On the complexity of models of arithmetic, this Journal, vol. 47 (1982), pp. 403415.
[19] RessayreJ.-P., Nonstandard universes with strong embeddings, and their finite approximations , Logic and Combinatorics, Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1987, pp. 333358.
[20] RogersH., Theory of Recursive Functions and Effective Computability, McGraw-Hill, Cambridge, MA, 1967.
[21] ShenA., Are random axioms useful? Math. ArXiv, http://arxiv.org/pdf/1109.5526.pdf.
[22] SimpsonS., Subsystems of Second Order Arithmetic, Springer-Verlag, Berlin, 1999.
[23] SmoryńskiC., Modal Logic and Self-reference, Springer-Verlag, Berlin, 1985.
[24] VerbruggeR. and VisserA., A small reflection principle for bounded arithmetic, this Journal, vol. 59 (1994), pp. 785812.
[25] WoodinW. H., A potential subtlety concerning the distinction between determinism and nondeterminism , Infinity, New Research Frontiers (HellerM. and WoodinW. H., 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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