Assume one has. for $V$ and for some transitive class $M$, an elementary embedding
$j$: $V$$\rightarrow$$M$ and that $j$$\neq$$id$, where $id$ is the identity.
If $V$ and $M$ satisfy $ZFC$ then the following Theorem holds
Thm. Let $\alpha$ be an ordinal.
(i) For every $\alpha$, $j$($\alpha$)$\ge$$\alpha$
(ii) $j$ moves some ordinal.
Let $\delta$ be the least ordinal moved by $j$. $\delta$ is called the critical point of $j$.
It can be proven in $ZFC$ that $\delta$ is always a cardinal.
Suppose now that $j$: $V$$\rightarrow$$M$, and $V$ and $M$ both satisfy $ZF$+$\lnot$$AC$. How can the critical point $\delta$ be defined when there exist incomparable cardinals?
I hope that this is not too silly a question. If it turns out to be silly, I will happily delete it.
(Addendum: Regarding Noah's answer to my question ("...even though the cardinalities of $V$ may not be well-ordered, the cardinals and ordinals definitely will be..."), in $ZF$+$\lnot$$AC$, there are cardinals which are not ordinals, and these may not be comparable. Since under a nontrivial elementary embedding no ordinal would be moved, could such cardinals be critical points of such an embedding?)