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$0^{\sharp}$ and elementary end extensions of $V_{\kappa}$


Author: Amir Leshem
Journal: Proc. Amer. Math. Soc. 129 (2001), 2445-2450
MSC (1991): Primary 03E45, 03E55
Published electronically: January 18, 2001
MathSciNet review: 1823930
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Abstract:

In this paper we prove that if $\kappa$ is a cardinal in $L[0^{\sharp}]$, then there is an inner model $M$ such that $M \models (V_{\kappa},\in)$ has no elementary end extension. In particular if $0^{\sharp}$ exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than $\aleph_1$ of uncountable cofinality in $L[0^{\sharp}]$ is Mahlo in every strict inner model of $L[0^{\sharp}]$.


References [Enhancements On Off] (What's this?)

  • 1. A. Beller, R. Jensen, and P. Welch, Coding the universe, London Mathematical Society Lecture Note Series, vol. 47, Cambridge University Press, Cambridge-New York, 1982. MR 645538
  • 2. Kenneth Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), no. 1, 65–76. MR 495118, 10.2307/2271949
  • 3. Jack H. Silver, Some applications of model theory in set theory, Ann. Math. Logic 3 (1971), no. 1, 45–110. MR 0409188
  • 4. M. C. Stanley, Backwards Easton forcing and 0^{♯}, J. Symbolic Logic 53 (1988), no. 3, 809–833. MR 961000, 10.2307/2274573
  • 5. A. Villaveces.
    Chains of elementary end extensions of models of set theory.
    Journal of Symbolic Logic, 63:1116-1136, September 1998. CMP 99:02
  • 6. A. Villaveces.
    Height of models of ZFC and the existence of end elementary extensions.
    Lecture Notes in Pure and Appl. Math., 203, 1999. CMP 99:12

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Additional Information

Amir Leshem
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem, Israel
Address at time of publication: Circuit and Systems, Faculty of Information Technology and Systems, Mekelweg 4, 2628CD Delft, The Netherlands
Email: leshem@cas.et.tudelft.nl

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05847-6
Keywords: Models of set theory, $0^{\sharp}$, inner models
Received by editor(s): October 19, 1999
Received by editor(s) in revised form: December 27, 1999
Published electronically: January 18, 2001
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2001 American Mathematical Society