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. p. 323

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 $\$, 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.

References

Large cardinals

{{settheory-stub