nLab club Berkeley cardinal

Idea

A large cardinal stronger than Reinhardt cardinals that is inconsistent with ZF with countable choice. In order to remove the inconsistency, one has to remove the axiom of countable choice to get bare ZF + a club Berkeley cardinal.

Definition

In ZF, a cardinal $\delta$ is a club Berkeley cardinal if it is regular and, for every club set (closed and unbounded set) $C \subseteq \delta$ and every transitive set $M$ containing $\delta$, there exists a non-trivial elementary embedding $j: M \to M$ such that its critical point lands exactly inside that club: $\mathrm{crit}(j) \in C$.

Related concepts

• large cardinal

• elementary embedding

• Berkeley cardinal

• limit club Berkeley cardinal

References

• Joan Bagaria, Peter Koellner, W. Hugh Woodin: Large Cardinals Beyond Choice, The Bulletin of Symbolic Logic, Vol. 25, No. 3 (September 2019), pp. 283-318 (36 pages) [jstor:stable/26788522]