Extender model: Difference between revisions

Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]]<nowiki> cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is [[ordinal-definable]]. In fact, an even stronger theorem holds: assume the HOD hypothesis holds and there is an [[extendible]] cardinal. Then there is an ordinal \(\lambda\) so that, for all \(\gamma > \lambda\), if \(j: \mathrm{HOD} \cap V_{\gamma+1} \to \mathrm{HOD} \cap V_{j(\gamma)+1}\) is an elementary embedding with \(j(\lambda) = \lambda\) (so the critical point is either above or far below \(\lambda\)), then \(j\) is hereditarily ordinal-definable.