Extender (set theory)
A (κ, λ)-extender can be defined as an elementary embedding of some model M of ZFC− (ZFC minus the power set axiom) having critical point κ ε M, and which maps κ to an ordinal at least equal to λ. It can also be defined as a collection of ultrafilters, one for each n-tuple drawn from λ.
Formal definition of an extender
Let κ and λ be cardinals with κ≤λ. Then, a set is called a (κ,λ)-extender if the following properties are satisfied:
- each Ea is a κ-complete nonprincipal ultrafilter on [κ]<ω and furthermore
- at least one Ea is not κ+-complete,
- for each , at least one Ea contains the set .
- (Coherence) The Ea are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).
- (Normality) If f is such that , then for some .
- (Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of the ultrapowers Ult(V,Ea)).
By coherence, one means that if a and b are finite subsets of λ such that b is a superset of a, then if X is an element of the ultrafilter Eb and one chooses the right way to project X down to a set of sequences of length |a|, then X is an element of Ea. More formally, for , where , and , where m≤n and for j≤m the ij are pairwise distinct and at most n, we define the projection .
Then Ea and Eb cohere if
Defining an extender from an elementary embedding
One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
- Jech, Thomas (2002). Set Theory (3rd ed.). Springer. ISBN 3-540-44085-2.
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