nLab 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 Berkeley cardinal.

Definition

In ZF, a Berkeley cardinal $\delta$ is defined as a cardinal such that for every transitive set $M$ containing $\delta$ and every ordinal $\alpha \lt \delta$, there exists a non-trivial elementary embedding $j:M \to M$ with $\alpha \lt \mathrm{crit}(j) \lt \delta$. Berkeley cardinals in ZF are one of the largest large cardinal axioms possible.

Related concepts

• large cardinal

• elementary embedding

• Reinhardt cardinal

• 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]

• Marwan Salam Mohammd, Berkeley Cardinals and Vopěnka’s Principle [arXiv:2404.10455]

• Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals? MathOverflow (web)

• Wikipedia, Berkeley cardinal