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

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.