Why does a nontrivial $V \to V$ have a critical point?

Question

Let $V$ denote the von Neumann universe, and let $j: V \to V$ be a nontrivial (non-identity) elementary embedding. The critical point is the smallest $\kappa$ such that $j(\kappa) > \kappa$. The linked Wikipedia above indicates that any $j \ne \operatorname{id}$ has such a critical point; how might one show this?

Answer 1

First of all, by elementarity $\operatorname{rank}(j(x))=j(\operatorname{rank}(x))$. So it suffices to show that there is some $x$ whose rank is moved.

Now we can prove the following by induction:

Suppose that $\operatorname{rank}(j(x))=\operatorname{rank}(x)$ and for all $y$ such that $\operatorname{rank}(y)<\operatorname{rank}(x)$, $j(y)=y$, then $j(x)=x$.

Proof. Simply note that $y\in x\leftrightarrow j(y)\in j(x)\leftrightarrow y\in j(x)$. $\square$

What follows is that if $j$ was the identity up to rank $\alpha$, it will either move some element of rank $\alpha$ to be of a higher rank, or it will be the identity on elements of rank $\alpha$ as well. Therefore if $j$ is not the identity, there is some element moved to a higher rank, and that rank is an ordinal moved up. The least such ordinal is called the critical point.