8 1. ELEMENTAR Y EMBEDDING S PROOF O F THEORE M 1.1.5 . Assum e towards a contradiction tha t V = L an d tha t K is th e leas t measurabl e cardina l i n L. The n ther e i s a n el - ementary embeddin g j : V M induce d b y a ^-complet e ultrafilte r U o n K. Sinc e V = L , M mus t b e L also . B y elementarity , J(K) i s th e leas t measurable cardina l i n M , bu t thi s give s a contradiction, sinc e M = L , « is the leas t measurabl e cardina l i n L , an d ft J(K). D As a second proof, w e present a property o f measurable cardinal s whic h will b e ke y t o man y argument s i n thi s book . Firs t w e not e a genera l fac t about elementar y submodels . LEMMA 1.1.18 . Let M be a transitive model of ZFC - Replacement. Let t G X be sets in M and let Z an elementary submodel of M with X G Z. Assume that one of the following holds. Ord C M, M is a set and cof{M n Ord) \X\, Z H Ord is cofinal in M H Ord. Then Z[t] = {f(t) | / : I - M A / G 2 } is an elementary submodel of M. PROOF. W e sho w tha t Z[t] satisfie s th e Tarski-Vaugh t criterion . Fi x a formula (j) an d x\, ..., x n G Z[t] an d suppos e tha t M | = 3y0(y,xi,...,x n ). Let / i , . . . , f n b e function s i n Z wit h domai n X suc h tha t fi{t) X{ holds for eac h i. I f Ord C M o r \X\ cof(M n Ord) , the n ther e i s a n 7 7 G M such tha t fo r eac h s G X, i f M h 3 # ( y , / i ( S ) , . . , / ( S ) ) , then ther e exist s a y G M o f rank les s than7 7 suc h tha t Mh0(y,/i(«),---,/n(s)) , and s o there i s such a n rj in Z. Likewise , i f Z H Ord i s cofinal i n M H Ord then ther e exis t a n 7 7 G Z an d a y G M o f rank les s than7 7 suc h tha t In eithe r case , there exist s i n Z a wellordering o f the set s i n M o f rank les s than7 7 an d s o by Comprehensio n ther e i s a functio n g : X M i n Z suc h that fo r eac h 5 G X1 i f there exist s a y o f rank les s than7 7 suc h tha t then MN^sWJi W /»(«)) The existence of such a g suffices t o show that Z[t] \= 3y0(y , #i,..., xn). D 1.1.19 Remark . I n ou r application s o f Lemm a 1.1.18 , th e structure s M will typicall y b e set s o f th e for m V 1 . Anothe r versio n o f th e lemm a hold s for set s o f the for m #(7 ) fo r regula r 7 , withou t th e assumptio n tha t H(j) satisfies th e Powerse t Axiom .
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