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 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 . 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.

References

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  1. ^ Jech, Thomas (2002). Set Theory. Berlin: Springer-Verlag. ISBN 3-540-44085-2. p. 323