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And Elementary End Extensions of V Κ

In this paper we prove that if κ is a cardinal in L[0]], then there is an inner model M such that M |= (Vκ,∈) has no elementary end extension. In particular if 0] exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than א1 of uncountable cofinality in L[0]] is Mahlo in every strict inner model of L[0]].