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.

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:

By coherence, one means that if a and b are finite subsets of λ 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 ${\displaystyle b=\{\alpha _{1},\dots ,\alpha _{n}\}}$, where ${\displaystyle \alpha _{1}<\dots <\alpha _{n}<\lambda }$, and ${\displaystyle a=\{\alpha _{i_{1}},\dots ,\alpha _{i_{m}}\}}$, where m≤n and for j≤m the i_j are pairwise distinct and at most n, we define the projection ${\displaystyle \pi _{ba}:\{\xi _{1},\dots ,\xi _{n}\}\mapsto \{\xi _{i_{1}},\dots ,\xi _{i_{m}}\}\ (\xi _{1}<\dots <\xi _{n})}$.

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