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

If ${\displaystyle N}${\displaystyle N} is V , then ${\displaystyle \kappa }${\displaystyle \kappa } (the critical point of ${\displaystyle j}${\displaystyle j}) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a ${\displaystyle \kappa }${\displaystyle \kappa }-complete, non-principal ultrafilter over ${\displaystyle \kappa }${\displaystyle \kappa }. Specifically, one may take the filter to be ${\displaystyle \{A\mid A\subseteq \kappa \land \kappa \in j(A)\}}${\displaystyle \{A\mid A\subseteq \kappa \land \kappa \in j(A)\}}, which defines a bijection between elementary embeddings and ultrafilters.^[2] Generally, there will be many other <κ-complete, non-principal ultrafilters over ${\displaystyle \kappa }${\displaystyle \kappa }. However, ${\displaystyle j}${\displaystyle j} might be different from the ultrapower(s) arising from such filter(s).

If ${\displaystyle N}${\displaystyle N} and ${\displaystyle M}${\displaystyle M} are the same and ${\displaystyle j}${\displaystyle j} is the identity function on ${\displaystyle N}${\displaystyle N}, then ${\displaystyle j}${\displaystyle j} is called "trivial". If the transitive class ${\displaystyle N}${\displaystyle N} is an inner model of ZFC and ${\displaystyle j}${\displaystyle j} has no critical point, i.e. every ordinal maps to itself, then ${\displaystyle j}${\displaystyle j} is trivial.^[2]

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