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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 number|ordinal]] which is not mapped to itself.<ref>{{cite book | last = Jech | first = Thomas | title = Set Theory | publisher = Springer-Verlag | location = Berlin | year = 2002 | isbn = 3-540-44085-2 }} p. 323</ref> | |||
Suppose that j : ''N'' → ''M'' is an elementary embedding where ''N'' and ''M'' are transitive classes and j is definable in ''N'' by a formula of set theory with parameters from ''N''. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(ω)=ω. If j(α)=α for all α<κ and j(κ)>κ, then κ is said to be the critical point of j. | |||
If ''N'' is ''[[Von Neumann universe|V]]'', then κ (the critical point of j) is always a [[measurable cardinal]], i.e. an uncountable [[cardinal number]] κ such that there exists a <κ-complete, non-principal [[ultrafilter]] over κ. Specifically, one may take the filter to be <math> \{A \vert A \subseteq \kappa \land \kappa \in j (A) \} \,.</math> Generally, there will be many other <κ-complete, non-principal ultrafilters over κ. However, j might be different from the ultrapower(s) arising from such filter(s). | |||
If ''N'' and ''M'' are the same and j is the identity function on ''N'', then j is called "trivial". If transitive class ''N'' is an [[inner model]] of ZFC and j has no critical point, i.e. every ordinal maps to itself, then j is trivial. | |||
==References== | |||
{{reflist}} | |||
{{DEFAULTSORT:Critical Point (Set Theory)}} | |||
[[Category:Large cardinals]] | |||
{{settheory-stub}} | |||
Latest revision as of 22:49, 26 February 2013
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 is an elementary embedding where N and M are transitive classes and j is definable in N by a formula of set theory with parameters from N. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(ω)=ω. If j(α)=α for all α<κ and j(κ)>κ, then κ is said to be the critical point of j.
If N is V, then κ (the critical point of j) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a <κ-complete, non-principal ultrafilter over κ. Specifically, one may take the filter to be Generally, there will be many other <κ-complete, non-principal ultrafilters over κ. However, j might be different from the ultrapower(s) arising from such filter(s).
If N and M are the same and j is the identity function on N, then j is called "trivial". If transitive class N is an inner model of ZFC and j has no critical point, i.e. every ordinal maps to itself, then j is trivial.
References
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