Von Neumann–Bernays–Gödel set theory

In axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal $\kappa$ is called Shelah iff for every $f:\kappa \to \kappa$, there exists a transitive class $N$ and an elementary embedding $j:V\to N$ 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.

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