1.1. ULTRAPOWER S 3 principal an d nonprincipal fo r measure s i n th e sam e wa y tha t w e us e the m for ultrafilters , an d us e th e term s measure an d ultrafilter interchangeably . The existenc e of a countably complet e nonprincipa l measur e i s equivalen t to th e existenc e o f a measurabl e cardinal . LEMM A 1.1.4 . Suppose that U is a countably complete nonprincipal ultra- filter on a set X 7 and let K be the completeness ofU. Then K is measurable. P R O O F . Le t A a (a K) b e set s in U such that H{A a : a K} 0 U. Sinc e the completenes s K of U i s infinite , b y addin g on e mor e se t i f necessar y w e may assum e tha t 0{A a : a K} = 0 . Sinc e U i s close d unde r intersection s of siz e /c , by replacin g eac h A a wit h K=n A 0 , (3a and the n thinnin g th e sequenc e i f necessary , w e ma y assum e tha t fo r eac h nonzero a n, A a i s a prope r subse t o f n{A ^ : (3 a}. Le t Bo = X \ Ao , and fo r eac h nonzer o a n le t B*= f]A p \A a . Then th e B a } s ar e pairwis e disjoint , eac h B a i s th e complemen t o f a se t i n [/, an d th e unio n o f th e B a 's i s X. Defin e a n ultrafilte r U* o n n b y lettin g each E C K be i n U* i f an d onl y i f U{B a : a G E} G U. The n sinc e U i s a ^-complete ultrafilter , [7 * must b e on e also . Theorems 1.1. 5 an d 1.1. 6 sho w that tw o of the earlies t consistenc y result s produced b y th e metho d o f forcin g ca n als o b e don e withou t forcing , fro m measurable cardinals . Th e first o f thes e wa s know n befor e th e discover y o f forcing, bu t th e secon d wa s discovere d afterwards . T H E O R E M 1.1.5 . (134]^ ) If there is a measurable cardinal then V ^ L. T H E O R E M 1.1.6 . ([24]) Assume that there are uncountably many mea- surable cardinals. Then the Axiom of Choice fails in L{Ord u ). The mode l L(Ord u ) ha s severa l equivalen t definitions , th e easies t o f which i s probabl y L{Ord")= | J L(V Ul (a)). a^Ord This model , first studie d i n [6] , i s know n a s th e Chan g model , an d i s th e smallest inne r mode l o f Z F containin g ever y countabl e sequenc e o f ordinal s in V . Before provin g Theorem s 1.1. 5 an d 1.1.6 , w e revie w som e basi c fact s about ultrapower s an d measurabl e cardinals . We sa y tha t a n embeddin g i s nontrivial i f i t i s no t th e identit y ma p o n its domain .
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