Extender model: Difference between revisions

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 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 have their fine structure analysed -, or are 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 [[Supercompact|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. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]].

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.

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 \(U \cap N \in N\).