1.1. ULTRAPOWER S 7 To su m up , a countabl y complet e nonprincipa l ultrafilte r induce s a nontrivial elementar y embeddin g o f V int o a n inne r mode l M. Th e com - pleteness o f th e ultrafilte r correspond s t o th e critica l poin t o f th e embed - ding. Furthermore , a nonprincipa l norma l ultrafilte r U o n a n uncount - able cardina l K induce s a n elementar y embeddin g j : V M suc h tha t U = {x CK\ Kej(X)}. The followin g fac t abou t measurabl e cardinal s wil l b e use d i n Chapte r 3. 1.1.14 Exercise . Suppos e tha t K is a measurabl e cardina l an d le t fi b e a normal measur e o n K. Le t A b e a se t i n \i an d le t / : A V K b e a functio n such tha t fo r eac h a i n A , f(a) i s a closed unbounde d subse t o f a. Sho w tha t there i s a clu b C C K such tha t fo r al l 7 i n C , th e se t {a G A | 7 G / ( a )} i s in /i . (Hint : Conside r J ( / ) ( K ) , wher e j i s th e embeddin g derive d fro m /1. ) 1.1.15 Remark . Measurabl e cardinal s ar e preserve d unde r smal l forcing . That is , i f K is a measurabl e cardina l an d P i s a partia l orde r i n V K1 the n K i s stil l measurabl e afte r forcin g wit h P . Th e sam e fac t hold s fo r al l th e large cardinal s considere d i n thi s book , i n particula r fo r strongl y inaccessibl e cardinals, Woodi n cardinals , strongl y compac t cardinal s an d supercompac t cardinals. Fo r measurabl e cardinal s th e proo f i s largel y th e sam e a s th e proof o f th e followin g lemma , whic h wil l b e use d i n th e nex t section . LEMM A 1.1.16 . Suppose that \i is a nonprincipal ^-complete ultrafilter on a set X and that P is partial order in V K . Let G C P be a V-generic filter, and suppose that F is a function in V[G] from X to the ordinals. Then there is a function F * : X Ord in V and a set A G V(X)V D \i such that F(a) = F*(a) for all a G A. PROOF. Workin g i n V, us e genericity . Le t r b e a P-nam e fo r F an d fi x po G P. Choos e function s n : X P an d F* : X Ord suc h tha t fo r al l a G X, 7r(a ) po an d 7r(a)lhr(a) = F*(a) . Since P G VKl ther e mus t exis t a qo G P suc h tha t {a G X | n(a) = qo} G /JL. Let A = {a G X \ 7r(a) = 40} . The n g 0 I^Va G A F(a) = F*(a) . 1.1.17 Exercise . Suppos e tha t K is a measurabl e cardinal , an d le t P b e a forcing constructio n i n V K . I f \i i s a ^-complet e measur e i n V o n a se t X and G c P i s l/-generic , the n i n V[G] w e defin e th e canonical extension /i + of / / t o b e th e se t o f A C X suc h tha t A D B fo r som e Be.^. Sho w tha t /i + i s Av-complete , an d tha t ever y ^-complet e measur e o n X i n V[G] i s th e canonical extensio n o f a measur e i n V. We no w prov e Theorem s 1.1. 5 an d 1.1. 6
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