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:

1. each $E_a$ 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 $\{s\in [\kappa {]}^{|a|}:\alpha \in s\}.$
2. (Coherence) The $E_a$ are coherent (so that the ultrapowers Ult(V,E_a) 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 $$\begin{gathered} b\supseteq a, \\ \{t\in {\kappa }^{|b|}:(f\circ {\pi }_{ba})(t)\in t\}\in E_b. \end{gathered}$$
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 $a$ 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 $m\le n$ and for $j\le m$ the $i_j$ are pairwise distinct and at most $n,$ we define the projection $$\begin{gathered} {\pi }_{ba}:\{{\xi }_1,\dots ,{\xi }_n\}\mapsto \{{\xi }_{i_1},\dots ,{\xi }_{i_m}\} \\ ({\xi }_1<\dots <{\xi }_n). \end{gathered}$$ Then $E_a$ and $E_b$ cohere if $$X\in E_a⟺\{s:{\pi }_{ba}(s)\in X\}\in E_b.$$

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: $$\text{for }a\in [\lambda {]}^{<\omega },X\subseteq [\kappa {]}^{<\omega }:X\in E_a⟺a\in j(X).$$ One can then show that $E$ has all the properties stated above in the definition and therefore is a (κ,λ)-extender.

References

• 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.