Extender model: Difference between revisions
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Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are
In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - 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 \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability.
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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