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

If  N {\displaystyle N} is V, then  κ {\displaystyle \kappa } (the critical point of  j {\displaystyle j} ) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a  κ {\displaystyle \kappa } -complete, non-principal ultrafilter over  κ {\displaystyle \kappa } . Specifically, one may take the filter to be  { A ∣ A ⊆ κ ∧ κ ∈ 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  κ {\displaystyle \kappa } . However,  j {\displaystyle j} might be different from the ultrapower(s) arising from such filter(s).

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

## References

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1. ^ Jech, Thomas (2002). Set Theory. Berlin: Springer-Verlag. ISBN 3-540-44085-2. p. 323
2. ^ a b Neeman, Itay (2010). "Ultrafilters and large cardinals". Ultrafilters across Mathematics. AMS.

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