Extender model

Extender models are inner models, which have similar fine structure to Gödel's \(L\), but which are able to accommodate large cardinals, typically at the level of measurable cardinals and above. Extender models are typically either constructed - where they typically have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) and their fine structure analysed - or defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for supercompactness has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out.

In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an 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 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, if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability.

For example, "\(N\) is a weak extender model for \(\kappa\)'s 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 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 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 \(E \cap N \in N\).