1.1. ULTRAPOWER S 13 by th e choic e o f h a . Thi s verifie s tha t n induce s a n orde r preservin g bijectio n 7f : Ordj°(Xl)/jo(in) - Ord Xl /^ Putting everythin g together , w e ge t finally tha t j \ \Ord = \Ord. T o see this , not e tha t fo r eac h ordina l 7 , i f g i s th e constan t functio n fro m X\ t o {7} , the n [g] /xi = ji(7) . Likewise , i f / i s th e constan t functio n fro m jo(^i ) t o {7} , then [/] Jo(m ) = j?(7) - Bu t ix(g) = / , an d sinc e TT induce s th e isomorphism 7f , thes e tw o value s ar e th e same . 1.1.26 Remark . Lemm a 1.1.2 5 ca n fai l i f w e dro p th e requiremen t tha t Xo E V Kl , i.e. , tha t |Xo | i s les s tha n th e completenes s o f /xi . LEMM A 1.1.27 . Given an ordinal 7 , let A 1 be the class of K such that there exists a countably complete nonprincipal ultrafilter \i of completeness K such that j (7) ^ 7 , for j : V " M t/i e embedding induced by fi. Then for all ordinals 7 , ^4 7 i s finite. P R O O F . Suppos e tha t th e lemm a fails , an d le t 7 b e th e leas t ordina l p such tha t A p i s infinite . Fi x th e followin g object s (fii : i CJ) , a sequenc e o f countabl y complet e nonprincipa l ultra - filters, a n increasin g sequenc e o f cardinal s (KI : i u) suc h tha t eac h Ki is the completenes s o f \i{ fo r eac h i a , th e elementar y embedddin g ji : V Mi give n b y Mi, and suppos e tha t fo r eac h i u, ^(7 ) 7 ^ 7. By Lemm a 1.1.25 , fo r eac h intege r k 0 , j£[Or i = j^fOrd , wher e j £ i s the embeddin g compute d i n MQ from jo(Mfc) - S o M 0 | = V f c a \ { 0 } j g ( 7 ) ^ 7 - In Mo , Jo(7) i s the leas t p such tha t A p i s infinite, whic h mean s tha t jo(7 ) 7. Sinc e jo(7) canno t b e les s tha n 7 , jo(7 ) = 7 which give s a contradiction . PROOF O F T H E O R E M 1.1.6 . Le t (/c a : a UJ{) b e th e first a i-man y measurable cardinals . Fo r eac h a , le t \i a b e a tt a -complete nonprincipa l ultrafilter o n AC Q , and le t j Q : V M a b e th e induce d embedding . Le t A be the supremu m o f th e « a 's. W e hav e tha t A w C L(Ord u ). Suppos e toward s a contradictio n tha t ther e i s a wellorderin g n o f A ^ i n LiOrd?). The n TT i s ^-definable i n L{Ord u ) usin g a countabl e se t s o f ordinal s a s a parameter . Fix a regular cardina l 7 max{A , ran/c(7r), sup(s)} an d not e tha t j a ( 7 ) = 7 for al l a . B y makin g 7 sufficientl y large , an d possibl y addin g t o s , w e ca n assume tha t TT i s definabl e i n l/ 7 n LiOrd^) fro m 5 . I n particular , w e ma y fix a formul a (j) suc h tha t TT = { a E A w x A ^ I V1 n L{Ord") | = 0(a , s)}.
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