Abstract
In this paper we prove that if κ \kappa is a cardinal in L [ 0 ♯ ] L[0^{\sharp }] , then there is an inner model M M such that M ⊨ ( V κ , ∈ ) M \models (V_{\kappa },\in ) has no elementary end extension. In particular if 0 ♯ 0^{\sharp } exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than ℵ 1 \aleph _1 of uncountable cofinality in L [ 0 ♯ ] L[0^{\sharp }] is Mahlo in every strict inner model of L [ 0 ♯ ] L[0^{\sharp }] .
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