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 is an elementary embedding where and are transitive classes and is definable in by a formula of set theory with parameters from. Then must take ordinals to ordinals and must be strictly increasing. Also. If for all and, then is said to be the critical point of.
If is V, then is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a -complete, non-principal ultrafilter over. Specifically, one may take the filter to be. Generally, there will be many other <κ-complete, non-principal ultrafilters over. However, might be different from the ultrapower arising from such filter.
If and are the same and is the identity function on, then is called "trivial". If transitive class is an inner model of ZFC and has no critical point, i.e. every ordinal maps to itself, then is trivial.
