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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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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 $L[0^{\sharp}]$.

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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

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