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.

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 ${\displaystyle E_{a}}$  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 ${\displaystyle m\leq n}$  and for ${\displaystyle j\leq m}$  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

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

 

Defining an extender from an elementary embedding

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:

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

 

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

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.