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 -extender can be defined as an elementary embedding of some model of ZFC− having critical point, and which maps to an ordinal at least equal to. It can also be defined as a collection of ultrafilters, one for each -tuple drawn from.

Formal definition of an extender

Let κ and λ be cardinals with κ≤λ. Then, a set is called a -extender if the following properties are satisfied:
  1. each is a κ-complete nonprincipal ultrafilter on <ω and furthermore
  2. # at least one is not κ+-complete,
  3. # for each at least one contains the set
  4. The are coherent.
  5. If is such that then for some
  6. The limit ultrapower Ult is wellfounded is the direct limit of the ultrapowers Ult).
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, one defines as follows:
One can then show that has all the properties stated above in the definition and therefore is a -extender.