1.1. ULTRAPOWER S 9 1.1.20 R e m a r k . A sligh t modificatio n o f th e proo f o f Lemm a 1.1.1 8 show s that w e ca n replac e t i n th e statemen t o f th e lemm a wit h an y se t Y C X closed unde r finit e sequences , definin g th e elementar y submode l Z[Y] = U(y) \f:X^MAyEYAfeZ}. This fac t i s als o a corollar y o f th e lemm a itself . We no w appl y Lemm a 1.1.1 8 t o th e cas e wher e X i s a measurabl e car - dinal. LEMMA 1.1.21 . Let M be a transitive set model of ZFC - Replacement, and let X = M D Ord. Let K be a cardinal in M and let /i C V(K) be such that M \= u ji is a normal uniform ultrafilter on ft." Let Z be an elementary substructure of M with \i G Z and Z n V{K) G ^tt(^(ft)) M - Assume that either cof(X) ft or Z n A is cofinal in A . For each 7 G ft let Z[y} = {f(j) | / : K - ^ M A / G Z } . Then the following hold. (1) For each 7 G K, Z[^} - M. (2) r\{Aa K\AZ /inz}^0 , (3) For each 7 G n{A C ft \ A G /i H Z], Zfy] f l 7 = Z 0 7 = Z n K. P R O O F . Th e firs t conclusio n i s just Lemm a 1.1.18 . Th e secon d follow s from th e fac t tha t ji i s ^-complet e i n M an d Z f l V(n) i s a se t i n M o f cardinality les s tha n K. Fo r th e thir d part , not e tha t sinc e K\ a G Z P i \i for al l a G Z n ft, 7 sup(Z f l ft). Thi s show s th e secon d equality . Fo r th e first, fi x a functio n / : ft —• » M i n Z . I f 7(7 ) i s a n ordina l les s tha n 7 , the n {a G ft I / ( a) a } , bein g i n Z , mus t b e i n /i . Sinc e \i i s normal , ther e is som e /3 such tha t {a G ft | / ( a ) = (3} E [JL. The n 7(7 ) = /? , an d sinc e Z - M, (5 must b e i n Z . Now, i f ft i s measurabl e an d A ft i s suc h tha t cof(X) ft, the n b y Lemma 1.1.21 , i f X i s a countabl e elementar y submode l o f V\ the n ther e exists a countabl e elementar y submode l Y o f V\ containin g X suc h tha t Y f l ft i s a prope r end-extensio n o f X D ft. The n an y suc h X ca n b e end - extended belo w ft an y countabl e numbe r o f times , whic h show s tha t fo r an y function f: UJ\ UJ\ there i s a countabl e elementar y submode l X - V\ such tha t o.t.(X f l ft) f(X f l c^i) . Thi s contradict s th e statemen t V=L, as follows . Suppos e tha t y = L , an d fo r eac h a u\, le t / ( a ) b e th e leas t (3 such tha t a i s countabl e i n L^ . Le t A be th e leas t cardina l abov e ft suc h that cof(X) ft an d (V\) L L\. Suppos e tha t X i s a countabl e elementar y submodel o f (Vx) L suc h tha t o.t.(X H A) f(X f l ui), an d le t a = X n u)\. Then th e transitiv e collaps e o f X i s i 0 .t.(xnA)5 a n ( i i f o.t.(X f l A ) / ( » ) , then a i s countabl e i n I 0 .£.(xnA)5 whic h i s impossibl e sinc e a i s uncountabl e in th e transitiv e collaps e o f X.
Previous Page Next Page