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 (ZFC minus the power set axiom) 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

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

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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

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  • 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.