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 E=\{E_a|a\in [\lambda]^{<\omega}\} 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 \alpha\in\kappa, at least one Ea contains the set \{s\in[\kappa]^{|a|}:\alpha\in s\}.
  2. (Coherence) The Ea are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).
  3. (Normality) If f is such that \{s\in[\kappa]^{|a|}: f(s)\in\max s\}\in E_a, then for some b\supseteq a,\ \{t\in\kappa^{|b|}:(f\circ \pi_{ba})(t)\in t\}\in E_b.
  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 b=\{\alpha_1,\dots,\alpha_n\}, where \alpha_1<\dots<\alpha_n<\lambda, and a=\{\alpha_{i_1},\dots,\alpha_{i_m}\}, where mn and for jm the ij are pairwise distinct and at most n, we define the projection \pi_{ba}:\{\xi_1,\dots,\xi_n\}\mapsto\{\xi_{i_1},\dots,\xi_{i_m}\}\ (\xi_1<\dots<\xi_n).

Then Ea and Eb cohere if

 X\in E_a\Leftrightarrow \{s: \pi_{ba}(s)\in X\}\in E_b.

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 E=\{E_a|a\in [\lambda]^{<\omega}\} as follows:

\text{for }a\in[\lambda]^{<\omega}, X\subseteq [\kappa]^{<\omega}:\quad X\in E_a\Leftrightarrow a\in j(X).

One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.


  • 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.