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 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

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 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 ${\displaystyle \alpha \in \kappa }$, at least one E_a contains the set ${\displaystyle \{s\in [\kappa ]^{|a|}:\alpha \in s\}}$.
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 ${\displaystyle \{s\in [\kappa ]^{|a|}:f(s)\in \max s\}\in E_{a}}$, 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 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 ${\displaystyle b=\{\alpha _{1},\dots ,\alpha _{n}\}}$, where ${\displaystyle \alpha _{1}<\dots <\alpha _{n}<\lambda }$, and ${\displaystyle a=\{\alpha _{i_{1}},\dots ,\alpha _{i_{m}}\}}$, where m≤n and for j≤m the i_j are pairwise distinct and at most 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 E_a and E_b cohere if

${\displaystyle X\in E_{a}\Leftrightarrow \{s:\pi _{ba}(s)\in X\}\in E_{b}}$.

Defining an extender from an elementary embedding

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

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

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

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.