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.^[1]

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

If ${\displaystyle N}$ is V, then ${\displaystyle \kappa }$ (the critical point of ${\displaystyle j}$) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a ${\displaystyle \kappa }$-complete, non-principal ultrafilter over ${\displaystyle \kappa }$. Specifically, one may take the filter to be ${\displaystyle \{A\mid A\subseteq \kappa \land \kappa \in j(A)\}}$, which defines a bijection between elementary embeddings and ultrafilters.^[2] Generally, there will be many other <κ-complete, non-principal ultrafilters over ${\displaystyle \kappa }$. However, ${\displaystyle j}$ might be different from the ultrapower(s) arising from such filter(s).

If ${\displaystyle N}$ and ${\displaystyle M}$ are the same and ${\displaystyle j}$ is the identity function on ${\displaystyle N}$, then ${\displaystyle j}$ is called "trivial". If the transitive class ${\displaystyle N}$ is an inner model of ZFC and ${\displaystyle j}$ has no critical point, i.e. every ordinal maps to itself, then ${\displaystyle j}$ is trivial.^[2]