Critical point (set theory) explained
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
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
(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
\{A\midA\subseteq\kappa\land\kappa\inj(A)\}
. Generally, there will be many other <
κ-complete, non-principal ultrafilters over
. However,
might be different from the
ultrapower(s) arising from such filter(s).
If
and
are the same and
is the identity function on
, then
is called "trivial". If the transitive class
is an
inner model of
ZFC and
has no critical point, i.e. every ordinal maps to itself, then
is trivial.
Notes and References
- Book: Jech, Thomas . Thomas Jech . Set Theory . Springer-Verlag . Berlin . 2002 . 3-540-44085-2 . p. 323