nLab Reinhardt cardinal

Idea

A large cardinal that is inconsistent with ZFC due to Kunen's inconsistency theorem. In order to remove the inconsistency, one can either choose to remove the axiom of choice to get ZF + a Reinhardt cardinal, or one can choose to remove the axiom of replacement to get ZC + a Reinhardt cardinal.

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.

If one tries to remove the axiom of infinity from the set theory, a Reinhardt cardinal is sufficient to prove the axiom of infinity.

Definition

In ZF

A Reinhardt cardinal in a model $V$ of ZF is a critical point of a non-trivial elementary embedding $j:V \to V$ of the model into itself. Reinhardt cardinals in ZF are one of the largest large cardinal axioms possible.

In IZF and CZF

A Reinhardt set in a model of IZF or CZF $V$ 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.

In BZC and ZC

In BZC and ZC, the definition of a Reinhardt cardinal is the same as the definition of a Reinhardt cardinal in ZF. However, the absence of the axiom of replacement makes a Reinhardt cardinal consistent with the axiom of choice.

Moreover, Reinhardt cardinals in BZC do not imply Mahlo cardinals in the absence of the axiom of full separation; they only imply the recursively Mahlo cardinals, which are like Mahlo cardinals but restricted to bounded $\Delta_0$-statements in the definition.

One only needs to add the $I_3$ axiom to a meta-theory in order to describe BZC + a Reinhardt cardinal.

In the presence of the axiom of choice, Reinhardt cardinals are inconsistent with Vopěnka's principle, since Vopěnka’s principle implies the axiom of replacement.

Related concepts

• large cardinal

• elementary embedding

• Berkeley cardinal

References

• Hanul Jeon: How strong is a Reinhardt set over extensions of CZF?, [arXiv:2101.07455]

• Rohan Srivastava: The Landscape of Large Cardinals [arXiv:2205.01787]

• 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]

• Hanul Jeon: Is BZC inconsistent with Reinhardt cardinals, Mathematics Stackexchange (web)

• Wikipedia, Reinhardt cardinal

On large cardinal axioms without the axiom of replacement:

• Large cardinals without replacement, MathOverflow (web)

On realizability models for Reinhardt cardinals:

• Laura Fontanella, Guillaume Geoffroy, Richard Matthews: Realizability Models for Large Cardinals, In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) [10.4230/LIPIcs.CSL.2024.28]