Zero sharp: Difference between revisions

* There is a proper class of nontrivial elementary embedding \(j: L \to L\), all with different critical points.

* For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\).

 

While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following:

* Chang's conjecture

* There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\).

* There is a Ramsey cardinal.