Critical point (set theory)

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself.

Suppose that $$j: N \to M$$ is an elementary embedding where $$N$$ and $$M$$ are transitive classes and $$j$$ is definable in $$N$$ by a formula of set theory with parameters from $$N$$. Then $$j$$ must take ordinals to ordinals and $$j$$ must be strictly increasing. Also $$j(\omega) = \omega$$. If $$j(\alpha) = \alpha$$ for all $$\alpha < \kappa$$ and $$j(\kappa) > \kappa$$, then $$\kappa$$ is said to be the critical point of $$j$$.

If $$N$$ is V, then $$\kappa$$ (the critical point of $$j$$) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a $$\kappa$$-complete, non-principal ultrafilter over $$\kappa$$. Specifically, one may take the filter to be $$ \{A \mid A \subseteq \kappa \land \kappa \in j(A)\}$$. Generally, there will be many other <κ-complete, non-principal ultrafilters over $$\kappa$$. However, $$j$$ might be different from the ultrapower(s) arising from such filter(s).

If $$N$$ and $$M$$ are the same and $$j$$ is the identity function on $$N$$, then $$j$$ is called "trivial". If the transitive class $$N$$ is an inner model of ZFC and $$j$$ has no critical point, i.e. every ordinal maps to itself, then $$j$$ is trivial.