Kunen's inconsistency: Difference between revisions
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((Kunen's proof may not be generalizable to j : V_(λ+1) -> V_(λ+1), but what if something else is?)) |
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* For all limit cardinals \(\lambda\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\). |
* For all limit cardinals \(\lambda\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\). |
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However, it is believed that it can not be generalized to show the nonexistence of a nontrivial elementary embedding \(j: V_{\lambda+1} \to V_{\lambda+1}\). Therefore, the rank-into-rank cardinals are believed to be |
However, it is believed that it can not be generalized to show the nonexistence of a nontrivial elementary embedding \(j: V_{\lambda+1} \to V_{\lambda+1}\). Therefore, the rank-into-rank cardinals are believed to be consistent, but barely teetering on the brink of inconsistency. |
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Kunen's inconsistency also implies that if \(\kappa\) is [[supercompact]], \(N\) is a [[Extender model|weak extender model]] for \(\kappa\)'s supercompactness, and \(j: N \to N\) is a nontrivial elementary embedding, then the critical point of \(j\) is less than \(\kappa\). It is known that there must be such an elementary embedding. |
Kunen's inconsistency also implies that if \(\kappa\) is [[supercompact]], \(N\) is a [[Extender model|weak extender model]] for \(\kappa\)'s supercompactness, and \(j: N \to N\) is a nontrivial elementary embedding, then the critical point of \(j\) is less than \(\kappa\). It is known that there must be such an elementary embedding. |
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