nLab Reinhardt cardinal

Idea

A large cardinal that is inconsistent with ZFC. In constructive set theory, one typically uses Reinhardt sets instead of Reinhardt cardinals, since cardinals are not well behaved in the absence of excluded middle.

One also has the notion of a super Reinhardt cardinal or a super Reinhardt set as generalizations of Reinhardt cardinals or Reinhardt sets.

Definition

A Reinhardt set in a model of set theory $V$ with an elementary embedding $J:V \to V$ is a inaccessible and transitive set $K$ such that $K \in J(K)$ and $j(x) = x$ for all $x \in K$.

Meanwhile there are multiple inequivalent definitions of a super Reinhardt set in constructive mathematics, which only coincide in the presence of excluded middle. See section 4.2 of Jeon & Matthews 2024 for more details.

Related concepts

• large cardinal

• CZF, IZF, ZFC

• elementary embedding

References

• Hanul Jeon, Richard Matthews, Very large set axioms over constructive set theories, The Bulletin of Symbolic Logic. 2024;30(4):455-535. [doi:10.1017/bsl.2024.8, arXiv:2204.05831]

• Wikipedia, Reinhardt cardinal