Extender (set theory)

Formal definition of an extender

Let κ and λ be cardinals with κ≤λ. Then, a set E = { E a | a ∈ [ λ ] < ω } is called a (κ,λ)-extender if the following properties are satisfied:

1. each E_a is a κ-complete nonprincipal ultrafilter on [κ]^<ω and furthermore
   1. at least one E_a is not κ⁺-complete,
   2. for each α ∈ κ , at least one E_a contains the set { s ∈ [ κ ] | a | : α ∈ s } .
2. (Coherence) The E_a are coherent (so that the ultrapowers Ult(V,E_a) form a directed system).
3. (Normality) If f is such that { s ∈ [ κ ] | a | : f ( s ) ∈ max s } ∈ E a , then for some b ⊇ a ,   { t ∈ κ | b | : ( f ∘ π b a ) ( t ) ∈ t } ∈ E b .
4. (Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of the ultrapowers Ult(V,E_a)).

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 E_b and one chooses the right way to project X down to a set of sequences of length |a|, then X is an element of E_a. More formally, for b = { α 1 , … , α n } , where α 1 < ⋯ < α n < λ , and a = { α i 1 , … , α i m } , where m≤n and for j≤m the i_j are pairwise distinct and at most n, we define the projection π b a : { ξ 1 , … , ξ n } ↦ { ξ i 1 , … , ξ i m }   ( ξ 1 < ⋯ < ξ n ) .

Then E_a and E_b cohere if

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 = { E a | a ∈ [ λ ] < ω } as follows:

One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.