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.
( κ , λ )
-extender can be defined as an elementary embedding of some model
of ZFC− (ZFC minus the power set axiom) having critical point
to an ordinal at least equal to
It can also be defined as a collection of ultrafilters, one for each
Let κ and λ be cardinals with κ≤λ.
is called a (κ,λ)-extender if the following properties are satisfied: By coherence, one means that if
are finite subsets of λ such that
is an element of the ultrafilter
and one chooses the right way to project
down to a set of sequences of length
α
α
α
α
α
α
are pairwise distinct and at most
we define the projection
ξ
ξ
ξ
ξ
Given an elementary embedding
which maps the set-theoretic universe
into a transitive inner model
with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines
has all the properties stated above in the definition and therefore is a (κ,λ)-extender.