Английская Википедия:Critical point (set theory)

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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 <math>j: N \to M</math> is an elementary embedding where <math>N</math> and <math>M</math> are transitive classes and <math>j</math> is definable in <math>N</math> by a formula of set theory with parameters from <math>N</math>. Then <math>j</math> must take ordinals to ordinals and <math>j</math> must be strictly increasing. Also <math>j(\omega) = \omega</math>. If <math>j(\alpha) = \alpha</math> for all <math>\alpha < \kappa</math> and <math>j(\kappa) > \kappa</math>, then <math>\kappa</math> is said to be the critical point of <math>j</math>.

If <math>N</math> is V, then <math>\kappa</math> (the critical point of <math>j</math>) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a <math>\kappa</math>-complete, non-principal ultrafilter over <math>\kappa</math>. Specifically, one may take the filter to be <math> \{A \mid A \subseteq \kappa \land \kappa \in j(A)\}</math>. Generally, there will be many other <κ-complete, non-principal ultrafilters over <math>\kappa</math>. However, <math>j</math> might be different from the ultrapower(s) arising from such filter(s).

If <math>N</math> and <math>M</math> are the same and <math>j</math> is the identity function on <math>N</math>, then <math>j</math> is called "trivial". If the transitive class <math>N</math> is an inner model of ZFC and <math>j</math> has no critical point, i.e. every ordinal maps to itself, then <math>j</math> is trivial.

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Шаблон:Settheory-stub