Extender (set theory) explained

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

E = \{E _{a} | a\in [ λ ] ^{<\omega} \} is called a (κ,λ)-extender if the following properties are satisfied:

1. each

E _{a} is a κ-complete nonprincipal ultrafilter on [&kappa;]^<ω and furthermore

1. 1. at least one

E _{a} is not κ⁺-complete,

1. 1. for each

\alpha \in \kappa, at least one E _{a} contains the set \{s \in [\kappa] ^{|a|} : \alpha \in s\}.

1. (Coherence) The

E _{a} are coherent (so that the ultrapowers Ult(V,E_a) form a directed system).

1. (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.}

1. (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 &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,...,\alpha} _{n\},} where \alpha _{1} < ... < \alpha _{n} < λ , and a = \{\alpha _{i _{1}} , ... ,\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,} ... , \xi _{n\}} \mapsto \{\xi _{i _{1}} , ... , \xi _{i _{m}} \} (\xi _{1} < ... < \xi _{n).}

Then

E _{a} and E _{b} cohere if$$X\; \backslash in\; E\_a\; \backslash iff\; \backslash \; \backslash in\; E\_b.$$

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 [ λ ] ^{<\omega} \} as follows:$$\backslash text\; a\; \backslash in\; [\backslash lambda]^,\; X\; \backslash subseteq\; [\backslash kappa]^\; :\; \backslash quad\; X\; \backslash in\; E\_a\; \backslash iff\; a\; \backslash in\; j(X).$$One can then show that E has all the properties stated above in the definition and therefore is a (κ,&lambda;)-extender.

References

• Book: Kanamori, Akihiro. Akihiro Kanamori

. Akihiro Kanamori. 2003. Springer. The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite. 2nd. 3-540-00384-3.

• Book: Jech, Thomas. Thomas Jech

. Thomas Jech. 2002. Springer. Set Theory. 3rd. 3-540-44085-2.