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.^[1]

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

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

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

References

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