Extender (set theory)

In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender. 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 E={Ea|a∈[λ]<ω} is called a (κ,λ)-extender if the following properties are satisfied:

1. each Ea is a κ-complete nonprincipal ultrafilter on [κ]<ω and furthermore
1. at least one Ea is not κ+-complete,
2. for each α∈κ, at least one Ea contains the set {s∈[κ]|a|:α∈s}.
2. (Coherence) The Ea are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).
3. (Normality) If f is such that {s∈[κ]|a|:f(s)∈max⁡s}∈Ea, then for some b⊇a, {t∈κ|b|:(f∘πba)(t)∈t}∈Eb.
4. (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 b={α1,…,αn}, where α1<…<αn<λ, and a={αi1,…,αim}, where m≤n and for j≤m the ij are pairwise distinct and at most n, we define the projection πba:{ξ1,…,ξn}↦{ξi1,…,ξim} (ξ1<…<ξn).

Then Ea and Eb cohere if X∈Ea⟺{s:πba(s)∈X}∈Eb.

## Defining an extender from an elementary embedding

Given an elementary embedding j:V->M, which maps the set-theoretic universe V into a transitive inner model M, with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines E={Ea|a∈[λ]<ω} as follows: for a∈[λ]<ω,X⊆[κ]<ω:X∈Ea⟺a∈j(X). One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.

## References

* 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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