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 $\{A\mid A\subseteq \kappa \wedge \kappa \in j(A)\}$, which defines a bijection between elementary embeddings and ultrafilters. 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.

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