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 -extender can be defined as an elementary embedding of some model M of ZFC− (ZFC minus the power set axiom) having critical point, and which maps \kappa to an ordinal at least equal to \lambda. It can also be defined as a collection of ultrafilters, one for each n-tuple drawn from \lambda.

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: 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 where and where m \leq n and for j \leq m the i_j are pairwise distinct and at most n, we define the projection Then E_a and E_b cohere if

Defining an extender from an elementary embedding

Given an elementary embedding 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 as follows: One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.