Shelah cardinal

In axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal $$\kappa$$ is called Shelah iff for every $$f:\kappa\rightarrow\kappa$$, there exists a transitive class $$N$$ and an elementary embedding $$j:V\rightarrow N$$ with critical point $$\kappa$$; and $$V_{j(f)(\kappa )}\subset N$$.

A Shelah cardinal has a normal ultrafilter containing the set of weakly hyper-Woodin cardinals below it.