Extendible: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "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...") |
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Extendible cardinals are a powerful large cardinal notion, which can be considered a significant strengthening of [[Weakly compact cardinal|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_{
* 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 \(j(\kappa) > \lambda\).
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