Related Directory Categories
- Set Theory (71)
- People (45)
Sources
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
Critical point (set theory)
From Wikipedia, the free encyclopedia
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.
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-additive, non-principal ultrafilter with base set κ.
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.
 |
This mathematical logic-related article is a stub. You can help Wikipedia by expanding it. |