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An inner model theoretic proof of Becker’s theorem

  • Grigor SargsyanEmail author
Article
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Abstract

We re-prove Becker’s theorem from Becker (Isr J Math 40(3–4):229–234, 1981) by showing that \(AD^{L({\mathbb {R}})}\) implies that \(L({\mathbb {R}})\vDash ``\omega _2\) is Open image in new window -supercompact”. Our proof uses inner model theoretic tools instead of Baire category. We also show that \(\omega _2\) is \(<\Theta \)-strongly compact.

Keywords

Set theory Inner model theory Directed systems Descriptive set theory 

Mathematics Subject Classification

03E15 03E35 03E45 03E55 03E57 03E60 

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References

  1. 1.
    Becker, H.: Determinacy implies that \(\aleph _{2}\) is supercompact. Isr. J. Math. 40(3–4), 229–234 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Becker, H., Jackson, S.: Supercompactness within the projective hierarchy. J. Symb. Logic 66(2), 658–672 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Harrington, L.A., Kechris, A.S.: Ordinal Games and their Applications. Logic, Methodology and Philosophy of Science VI. Studies in Logic and the Foundations of Mathematics (Hannover, 1979), vol. 104, pp. 273–277. North-Holland, Amsterdam (1982)Google Scholar
  4. 4.
    Jackson, S.: The weak square property. J. Symb. Logic 66(2), 640–657 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Neeman, I.: Inner models and ultrafilters in \(L(\mathbb{R})\). Bull. Symb. Logic 13(1), 31–53 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sargsyan, G.: \(AD_{\mathbb{R}}\) implies that sets of reals are \(\theta \)-uB (to appear). http://www.grigorsargis.net/. Accessed Dec 2019
  7. 7.
    Steel, J.R.: An outline of inner model theory. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1–3, pp. 1595–1684. Springer, Dordrecht (2010)CrossRefGoogle Scholar
  8. 8.
    Steel, J.R., Woodin, W.H.: HOD as a core model. In: Ordinal Definability and Recursion Theory: The Cabal Seminar, vol. III. Lecture Notes Logic , vol. 43, pp. 257–345. Association for Symbolic Logic and Cambridge University Press, Ithaca, NY (2016)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Rutgers The State University of New JerseyNew BrunswickUSA

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