Extendible
Extendible cardinals are a powerful large cardinal notion, which can be considered a significant strengthening of weakly compact cardinals, or a combination of supercompact and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations:
- For each \(\lambda > \kappa\), there is an elementary embedding \(j: V_{\lambda+1} \to V_{j(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\).
- For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\).
While the latter clearly characterization connects extendibles to supercompacts, the same can not be said for the first. However, this follows from Magidor's lemma: a cardinal \(\kappa\) is supercompact iff, for all \(\lambda > \kappa\), there exist \(\bar{\kappa} < \bar{\lambda} < \kappa\) and an elementary embedding \(j: V_{\bar{\lambda}+1} \to V_{\lambda+1}\) with critical point \(\bar{\kappa}\) so that \(j(\bar{\kappa}) = \kappa\). If \(N\) is a weak extender model for \(\kappa\)'s supercompactness, then, for all \(a \in V_\lambda\), the above characterisation of supercompactness holds and, for some \(\bar{a} \in V_{\bar{\lambda}}\):
- \(j(\bar{a}) = a\).
- \(j(N \cap V_{\bar{\lambda}}) = N \cap V_{\bar{\lambda}}\).
- \(j \upharpoonright (N \cap V_{\bar{\lambda}}) \in N\).