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Authors:	Amir Leshem

Affiliation:	Institute of Mathematics, Hebrew University, Jerusalem, Israel

Abstract:	In this paper we prove that if is a cardinal in $L[0^{sharp}]$ , then there is an inner model 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 of uncountable cofinality in $L[0^{sharp}]$ is Mahlo in every strict inner model of $L[0^{sharp}]$ .

Keywords:	Models of set theory $0^{sharp}$ inner models

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