Shelah cardinal

In axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal [math]\displaystyle{ \kappa }[/math] is called Shelah iff for every [math]\displaystyle{ f:\kappa\rightarrow\kappa }[/math], there exists a transitive class [math]\displaystyle{ N }[/math] and an elementary embedding [math]\displaystyle{ j:V\rightarrow N }[/math] with critical point [math]\displaystyle{ \kappa }[/math]; and [math]\displaystyle{ V_{j(f)(\kappa )}\subset N }[/math]. A Shelah cardinal has a normal ultrafilter containing the set of weakly hyper-Woodin cardinals below it.

References

• Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings of the American Mathematical Society 130/11, pp. 3385-3391, 2002, online