Extender (set theory)

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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 M 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 n-tuple drawn from λ.

Formal definition of an extender[edit]

Let κ and λ be cardinals with κ≤λ. Then, a set is called a (κ,λ)-extender if the following properties are satisfied:

  1. each Ea is a κ-complete nonprincipal ultrafilter on [κ] and furthermore
    1. at least one Ea is not κ+-complete,
    2. for each , at least one Ea contains the set .
  2. (Coherence) The Ea are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).
  3. (Normality) If f is such that , then for some .
  4. (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 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 mn and for jm 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[edit]

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(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.

References[edit]

  • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3. 
  • Jech, Thomas (2002). Set Theory (3rd ed.). Springer. ISBN 3-540-44085-2.