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 $(\kappa ,\lambda )$ -extender can be defined as an elementary embedding of some model $M$ of ZFC⁻ (ZFC minus the power set axiom) having critical point $\kappa \in M$ , and which maps $\kappa$ to an ordinal at least equal to $\lambda$ . It can also be defined as a collection of ultrafilters, one for each $n$ -tuple drawn from $\lambda$ .

Formal definition of an extender

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:

each $E_{a}$ $E_{a}$ is a κ-complete nonprincipal ultrafilter on [κ]^<ω and furthermore
1. at least one $E_{a}$ is not κ⁺-complete,
2. for each $\alpha \in \kappa ,$ at least one $E_{a}$ contains the set $\{s\in [\kappa ]^{|a|}:\alpha \in s\}.$
(Coherence) The $E_{a}$ are coherent (so that the ultrapowers Ult(V,E_a) form a directed system).
(Normality) If $f$ is such that $\{s\in [\kappa ]^{|a|}:f(s)\in \max s\}\in E_{a},$ then for some $b\supseteq a,\ \{t\in \kappa ^{|b|}:(f\circ \pi _{ba})(t)\in t\}\in E_{b}.$
(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 $a$ and $b$ are finite subsets of λ such that $b$ is a superset of $a,$ then if $X$ is an element of the ultrafilter $E_{b}$ and one chooses the right way to project $X$ down to a set of sequences of length $|a|,$ then $X$ is an element of $E_{a}.$ 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 $m\leq n$ and for $j\leq m$ the $i_{j}$ 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 $E_{a}$ and $E_{b}$ cohere if $X\in E_{a}\iff \{s:\pi _{ba}(s)\in X\}\in E_{b}.$

Defining an extender from an elementary embedding

Given an elementary embedding $j:V\to M,$ 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: ${\text{for }}a\in [\lambda ]^{<\omega },X\subseteq [\kappa ]^{<\omega }:\quad X\in E_{a}\iff a\in j(X).$ One can then show that $E$ has all the properties stated above in the definition and therefore is a (κ,λ)-extender.