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 &kappa; and &lambda; be cardinals with &kappa;&le;&lambda;. 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 &kappa;-complete nonprincipal ultrafilter on [&kappa;]&lt;&omega; and furthermore
 * 2) at least one $$E_a$$ is not &kappa;+-complete,
 * 3) for each $$\alpha \in \kappa,$$ at least one $$E_a$$ contains the set $$\{s \in [\kappa]^{|a|} : \alpha \in s\}.$$
 * 4) (Coherence) The $$E_a$$ are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).
 * 5) (Normality) If $$f$$ is such that $$\{s \in [\kappa]^{|a|}: f(s) \in \max s\} \in E_a,$$ then for some $$b \supseteq a,\ \{t \in \kappa^{|b|} : (f \circ \pi_{ba})(t) \in t\} \in E_b.$$
 * 6) (Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of the ultrapowers Ult(V,Ea)).

By coherence, one means that if $$a$$ and $$b$$ are finite subsets of &lambda; 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 $$m \leq n$$ and for $$j \leq m$$ the $$i_j$$ are pairwise distinct and at most $$n,$$ we define the projection $$\pi_{ba} : \{\xi_1, \dots, \xi_n\} \mapsto \{\xi_{i_1}, \dots, \xi_{i_m}\}\ (\xi_1 < \dots < \xi_n).$$

Then $$E_a$$ and $$E_b$$ cohere if $$X \in E_a \iff \{s : \pi_{ba}(s) \in X\} \in E_b.$$

Defining an extender from an elementary embedding
Given an elementary embedding $$j : V \to M,$$ 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: $$\text{for } a \in [\lambda]^{<\omega}, X \subseteq [\kappa]^{<\omega} : \quad X \in E_a \iff a \in j(X).$$ One can then show that $$E$$ has all the properties stated above in the definition and therefore is a (&kappa;,&lambda;)-extender.