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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.

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 $V_\kappa\prec V$ ($\mathrm{WR}(\kappa)$. Existence of
$\kappa$ is Weak Reinhardt Axiom ($\mathrm{WRA}$) by
Woodin).(Corazza, 2010):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 $V_\kappa\prec V_\lambda\prec V_\gamma$ (for some
$\gamma > \lambda >
\kappa$).(Baaz et al., 2011):(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.(Bagaria, 2017)

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.(Bagaria, 2017)

For a proper class $A$, cardinal $\kappa$ is called **$A$-super
Reinhardt** if for all ordinals $\lambda$ there is a non-trivial
elementary embedding $j : V \rightarrow V$ such that $\mathrm{crit}(j)
= \kappa$, $j(\kappa)\gt\lambda$ and $j^+(A)=A$. (where $j^+(A) :=
\cup_{α∈\mathrm{Ord}} j(A ∩
V_α)$)(Bagaria, 2017)

A **totally Reinhardt** cardinal is a cardinal $\kappa$ such that for
each $A ∈ V_{κ+1}$, $(V_\kappa, V_{\kappa+1})\vDash
\mathrm{ZF}_2 + \text{“There is an $A$-super Reinhardt
cardinal”}$.(Bagaria, 2017)

Totally Reinhardt cardinals are the ultimate conclusion of the Vopěnka hierarchy. A cardinal is Vopěnka if and only if, for every $A\subseteq V_\kappa$, there is some $\alpha\lt\kappa$ $\eta-$extendible for $A$ for every \(\eta\lt\kappa\), in that the witnessing embeddings fix $A\cap V_\zeta$. In its original conception Reinhardt cardinals were thought of as ultimate extendible cardinals, because if $j: V\rightarrow V$ is elementary, then so is $j\restriction V_{\kappa+\eta}: V_{\kappa+\eta}\rightarrow V_{j(\kappa+\eta)}$. It is as if one embedding works for all $\eta$.

$\mathrm{WRA}$ (1) implies thet there are arbitrary large $I1$ and super $n$-huge cardinals. Kunen inconsistency does not apply to it. It is not known to imply $I0$.(Corazza, 2010)

$\mathrm{WRA}$ (1) does not need $j$ in the language. It however 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.(Corazza, 2010)

$\mathrm{WR}(\kappa)$ (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.(Corazza, 2010)

Weakly Reinhardt cardinal(2) is inconsistent with $\mathrm{ZFC}$. $\mathrm{ZF} + \text{“There is a weakly Reinhardt cardinal(2)”}\rightarrow\mathrm{Con}(\mathrm{ZFC} + \text{“There is a proper class of $\omega$-huge cardinals”})$ (At least here $\omega$-huge=$I1$) (Woodin, 2009). You can get this by seeing that $V_\gamma\vDash\forall\alpha\lt\lambda(\exists\kappa’\gt\alpha(I1(\kappa’)\land\kappa’\lt\lambda))$.

If $\kappa$ is super Reinhardt, then there exists $\gamma\lt\kappa$ such that $(V_\gamma , V_{\gamma+1})\vDash \mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal”}$.(Bagaria, 2017)

If $\delta_0$ is the least Berkeley cardinal, then there is $\gamma\lt\delta_0$ such that $(V_\gamma , V_{\gamma+1})\vDash\mathrm{ZF}_2+\text{“There is a Reinhardt cardinal witnessed by $j$ and an $\omega$-huge above $\kappa_\omega(j)”$}$. (Here $\omega-$huge means $I3$). (Bagaria, 2017) Each club Berkeley cardinal is totally Reinhardt.(Bagaria, 2017)

- Corazza, P. (2010). The Axiom of Infinity and transformations j: V \to V.
*Bulletin of Symbolic Logic*,*16*(1), 37–84. https://doi.org/10.2178/bsl/1264433797 - Baaz, M., Papadimitriou, C. H., Putnam, H. W., Scott, D. S., & Harper, C. L. (2011).
*Kurt Gödel and the Foundations of Mathematics: Horizons of Truth*. Cambridge University Press. https://books.google.pl/books?id=Tg0WXU5_8EgC - Bagaria, J. (2017).
*Large Cardinals beyond Choice*. https://events.math.unipd.it/aila2017/sites/default/files/BAGARIA.pdf