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The Axiom of Strong Condensation, first introduced by Woodin in [14], is an abstract version of the Condensation Lemma of L. In this paper, we construct a set-sized forcing to obtain Strong Condensation for H(ω2). As an application, we show that “ZFC + Axiom of Strong Condensation + ”is consistent, which answers a question in [14]. As another application, we give a partial answer to a question of Jech by proving that “ZFC + there is a supercompact cardinal + any ideal on ω1 which is definable over H(ω2) is not precipitous” is consistent under sufficient large cardinal assumptions.



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[13]Wu, L., Sharp for the model of strong condensation, in preparation
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