Extender model: Difference between revisions
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For example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\). Also, "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\).
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 \(E \cap N \in N\).
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