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 either constructed - where they 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]].

 

 

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