Shelah cardinal

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

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