# Reinhardt

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The existence of Reinhardt cardinals has been refuted in $\text{ZFC}_2$ and $\text{GBC}$ by Kunen (Kunen inconsistency), the term is used in the $\text{ZF}_2$ context, although some mathematicians suspect that they are inconsistent even there.

## Definitions

A weakly Reinhardt cardinal(1) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ such that $V_\kappa ≺ V$ ($\mathrm{WR}(\kappa)$. Existence of $\kappa$ is Weak Reinhardt Axiom ($\mathrm{WRA}$) by Woodin).:p.58

A weakly Reinhardt cardinal(2) is the critical point $\kappa$ of a nontrivial elementary embedding $j:V_{\lambda+2}\to V_{\lambda+2}$ such that $V_\kappa ≺ V_\lambda ≺ V_\gamma$ (for some $\gamma > \lambda > \kappa$).:(definition 20.6, p. 455)

A Reinhardt cardinal is the critical point of a nontrivial elementary embedding $j:V\to V$ of the set-theoretic universe to itself.

A super Reinhardt cardinal $\kappa$, is a cardinal which is the critical point of elementary embeddings $j:V\to V$, with $j(\kappa)$ as large as desired.

For a proper class $A$, cardinal $κ$ is called $A$-super Reinhardt if for all ordinals $λ$ there is a non-trivial elementary embedding $j : V → V$ such that $\mathrm{crit}(j) = κ$, $j(κ) > λ$ and $j(A) = A$. (where $j(A) := ⋃_{α∈\mathrm{OR}} j(A ∩ V_α)$)

A totally Reinhardt cardinal is a cardinal $κ$ such that for each $A ∈ V_{κ+1}$, $\langle V_κ , V_{κ+1} \rangle \models \mathrm{ZF}_2 + \text{“There is an$A$-super Reinhardt cardinal”}$.

## Relations

• $\mathrm{WRA}$ (1) implies thet there are arbitrary large $\mathrm{I}_1$ and super $n$-huge cardinals. Kunen inconsistency does not apply to it. It is not known to imply $\mathrm{I}_0$.
• $\mathrm{WRA}$ (1) does not need $j$ in the language. It hovewer requires another extension to the language of $\mathrm{ZFC}$, because otherwise there would be no weakly Reinhardt cardinals in $V$ because there are no weakly Reinhardt cardinals in $V_\kappa$ (if $\kappa$ is the least weakly Reinhardt) — obvious contradiction.
• $\mathrm{WR}(\kappa)$ (1) implies that $\kappa$ is a measurable limit of supercompact cardinals and therefore is strongly compact. It is not known whether $\kappa$ must be supercompact itself. Requiring it to be extendible makes the theory stronger.
• Weakly Reinhardt cardinal(2) is inconsistent with $\mathrm{ZFC}$. $\mathrm{ZF} + \text{“There is a weakly Reinhardt cardinal(2)”} \implies \mathrm{Con}(\mathrm{ZFC} + \text{“There is a proper class of$\omega$-huge cardinals”})$ (at least here $\omega$-huge=I1) (Woodin, 2009).
• If $κ$ is super Reinhardt, then there exists $γ < κ$ such that $\langle V_γ , V_{γ+1} \rangle \models \mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal”}$.
• Totally Reinhardt cardinals are obviously stronger than super Reinhardt.
• If $δ_0$ is the least Berkeley cardinal, then there is $γ < δ_0$ s.t. $\langle V_γ , V_{γ+1} \rangle \models \mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal witnessed by$j$and an ω-huge above$κ_ω(j)”$}$.
• Each club Berkeley cardinal is totally Reinhardt.