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 [math]\displaystyle{ M }[/math] 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 [math]\displaystyle{ n }[/math]-tuple drawn from λ.

Formal definition of an extender

Let κ and λ be cardinals with κ≤λ. Then, a set [math]\displaystyle{ E = \{E_a | a\in [\lambda]^{\lt \omega}\} }[/math] is called a (κ,λ)-extender if the following properties are satisfied:

1. each [math]\displaystyle{ E_a }[/math] is a κ-complete nonprincipal ultrafilter on [κ]^<ω and furthermore
   1. at least one [math]\displaystyle{ E_a }[/math] is not κ⁺-complete,
   2. for each [math]\displaystyle{ \alpha \in \kappa, }[/math] at least one [math]\displaystyle{ E_a }[/math] contains the set [math]\displaystyle{ \{s \in [\kappa]^{|a|} : \alpha \in s\}. }[/math]
2. (Coherence) The [math]\displaystyle{ E_a }[/math] are coherent (so that the ultrapowers Ult(V,E_a) form a directed system).
3. (Normality) If [math]\displaystyle{ f }[/math] is such that [math]\displaystyle{ \{s \in [\kappa]^{|a|}: f(s) \in \max s\} \in E_a, }[/math] then for some [math]\displaystyle{ b \supseteq a,\ \{t \in \kappa^{|b|} : (f \circ \pi_{ba})(t) \in t\} \in E_b. }[/math]
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 [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are finite subsets of λ such that [math]\displaystyle{ b }[/math] is a superset of [math]\displaystyle{ a, }[/math] then if [math]\displaystyle{ X }[/math] is an element of the ultrafilter [math]\displaystyle{ E_b }[/math] and one chooses the right way to project [math]\displaystyle{ X }[/math] down to a set of sequences of length [math]\displaystyle{ |a|, }[/math] then [math]\displaystyle{ X }[/math] is an element of [math]\displaystyle{ E_a. }[/math] More formally, for [math]\displaystyle{ b = \{\alpha_1,\dots,\alpha_n\}, }[/math] where [math]\displaystyle{ \alpha_1 \lt \dots \lt \alpha_n \lt \lambda, }[/math] and [math]\displaystyle{ a = \{\alpha_{i_1},\dots,\alpha_{i_m}\}, }[/math] where [math]\displaystyle{ m \leq n }[/math] and for [math]\displaystyle{ j \leq m }[/math] the [math]\displaystyle{ i_j }[/math] are pairwise distinct and at most [math]\displaystyle{ n, }[/math] we define the projection [math]\displaystyle{ \pi_{ba} : \{\xi_1, \dots, \xi_n\} \mapsto \{\xi_{i_1}, \dots, \xi_{i_m}\}\ (\xi_1 \lt \dots \lt \xi_n). }[/math]

Then [math]\displaystyle{ E_a }[/math] and [math]\displaystyle{ E_b }[/math] cohere if [math]\displaystyle{ X \in E_a \iff \{s : \pi_{ba}(s) \in X\} \in E_b. }[/math]

Defining an extender from an elementary embedding

Given an elementary embedding [math]\displaystyle{ j : V \to M, }[/math] which maps the set-theoretic universe [math]\displaystyle{ V }[/math] into a transitive inner model [math]\displaystyle{ M, }[/math] with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines [math]\displaystyle{ E = \{E_a | a \in [\lambda]^{\lt \omega}\} }[/math] as follows: [math]\displaystyle{ \text{for } a \in [\lambda]^{\lt \omega}, X \subseteq [\kappa]^{\lt \omega} : \quad X \in E_a \iff a \in j(X). }[/math] One can then show that [math]\displaystyle{ E }[/math] 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.