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 (κ, λ)-extender can be defined as an elementary embedding of some model ${\displaystyle M}$ 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 ${\displaystyle n}$-tuple drawn from λ.

Let κ and λ be cardinals with κ≤λ. Then, a set ${\displaystyle E=\{E_a|a\in [\lambda {]}^{<\omega }\}}$ is called a (κ,λ)-extender if the following properties are satisfied:

1. each ${\displaystyle E_a}$ is a κ-complete nonprincipal ultrafilter on [κ]^<ω and furthermore
   1. at least one ${\displaystyle E_a}$ is not κ⁺-complete,
   2. for each ${\displaystyle \alpha \in \kappa ,}$ at least one ${\displaystyle E_a}$ contains the set ${\displaystyle \{s\in [\kappa {]}^{|a|}:\alpha \in s\}.}$
2. (Coherence) The ${\displaystyle E_a}$ are coherent (so that the ultrapowers Ult(V,E_a) form a directed system).
3. (Normality) If ${\displaystyle f}$ is such that ${\displaystyle \{s\in [\kappa {]}^{|a|}:f(s)\in \max s\}\in E_a,}$ then for some ${\displaystyle b\supseteq a,\{t\in {\kappa }^{|b|}:(f\circ {\pi }_{ba})(t)\in t\}\in E_b.}$
4. (Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of the ultrapowers Ult(V,E_a)).

By coherence, one means that if ${\displaystyle a}$ and ${\displaystyle b}$ are finite subsets of λ such that ${\displaystyle b}$ is a superset of ${\displaystyle a,}$ then if ${\displaystyle X}$ is an element of the ultrafilter ${\displaystyle E_b}$ and one chooses the right way to project ${\displaystyle X}$ down to a set of sequences of length ${\displaystyle |a|,}$ then ${\displaystyle X}$ is an element of ${\displaystyle E_a.}$ More formally, for ${\displaystyle b=\{{\alpha }_1,\dots ,{\alpha }_n\},}$ where ${\displaystyle {\alpha }_1<\dots <{\alpha }_n<\lambda ,}$ and ${\displaystyle a=\{{\alpha }_{i_1},\dots ,{\alpha }_{i_m}\},}$ where ${\displaystyle m\le n}$ and for ${\displaystyle j\le m}$ the ${\displaystyle i_j}$ are pairwise distinct and at most ${\displaystyle n,}$ we define the projection ${\displaystyle {\pi }_{ba}:\{{\xi }_1,\dots ,{\xi }_n\}\mapsto \{{\xi }_{i_1},\dots ,{\xi }_{i_m}\}({\xi }_1<\dots <{\xi }_n).}$

Then ${\displaystyle E_a}$ and ${\displaystyle E_b}$ cohere if $${\displaystyle X\in E_a⟺\{s:{\pi }_{ba}(s)\in X\}\in E_b.}$$

Given an elementary embedding ${\displaystyle j:V\to M,}$ which maps the set-theoretic universe ${\displaystyle V}$ into a transitive inner model ${\displaystyle M,}$ with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines ${\displaystyle E=\{E_a|a\in [\lambda {]}^{<\omega }\}}$ as follows: $${\displaystyle \text{for }a\in [\lambda {]}^{<\omega },X\subseteq [\kappa {]}^{<\omega }:X\in E_a⟺a\in j(X).}$$ One can then show that ${\displaystyle E}$ 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. 

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