Extender (set theory)
Introduction 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  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 -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  is a κ-complete nonprincipal ultrafilter on κ and furthermore## at least one  is not κ+-complete,## for each  at least one  contains the set # (Coherence) The  are coherent (so that the ultrapowers Ult(''V'',''Ea'') form a directed system).# (Normality) If  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  and  are finite subsets of λ such that  is a superset of  then if  is an element of the ultrafilter  and one chooses the right way to project  down to a set of sequences of length  then  is an element of  More formally, for  where  and  where  and for  the  are pairwise distinct and at most  we define the projection Then  and  cohere if
Defining an extender from an elementary embedding 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  as follows:One can then show that  has all the properties stated above in the definition and therefore is a (κ,λ)-extender.
References * *