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.
is an elementary embedding where
are transitive classes and
by a formula of set theory with parameters from
must be strictly increasing.
α < κ
is said to be the critical point of
) 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
, which defines a bijection between elementary embeddings and ultrafilters.
[2] Generally, there will be many other <κ-complete, non-principal ultrafilters over
might be different from the ultrapower(s) arising from such filter(s).
is the identity function on
is called "trivial".
is an inner model of ZFC and
has no critical point, i.e. every ordinal maps to itself, then
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