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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 is called a (κ,λ)-extender if the following properties are satisfied: # each ''Ea'' is a κ-complete nonprincipal ultrafilter on ()<ω and furthermore ## at least one ''Ea'' is not κ+-complete, ## for each , at least one ''Ea'' contains the set . # (Coherence) The ''Ea'' are coherent (so that the ultrapowers Ult(''V'',''Ea'') form a directed system). # (Normality) If ''f'' is such that , then for some . # (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 λ such that ''b'' is a superset of ''a'', then if ''X'' is an element of the ultrafilter ''Eb'' and one chooses the right way to project ''X'' down to a set of sequences of length |''a''|, then ''X'' is an element of ''Ea''. More formally, for , where , and , where ''m''≤''n'' and for ''j''≤''m'' the ''ij'' are pairwise distinct and at most ''n'', we define the projection . Then ''E''''a'' and ''Eb'' cohere if :. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Extender (set theory)」の詳細全文を読む スポンサード リンク
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