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Extender model: Difference between revisions
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Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\).
Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]]<nowiki> cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is
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