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 M of ZFC− 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 n-tuple drawn from λ.

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 Ea is a κ-complete nonprincipal ultrafilter on <ω and furthermore
  2. # at least one Ea is not κ+-complete,
  3. # for each, at least one Ea contains the set.
  4. The Ea are coherent.
  5. If f 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 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 Eb and one chooses the right way to project X down to a set of sequences of length |a|, then X is an element of Ea. More formally, for, where, and, where m≤n and for j≤m the ij are pairwise distinct and at most n, we define the projection.
Then Ea and Eb 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, one defines as follows:
One can then show that E has all the properties stated above in the definition and therefore is a -extender.