4 1. ELEMENTAR Y EMBEDDING S 1.1.7 Exercise . Sho w tha t i f M an d N ar e transitiv e model s o f ZF C an d j : M N i s a nontrivia l So-elementar y embeddin g (i.e. , preservin g th e ^-relation) wit h N C M , the n j i s no t th e identit y o n th e ordinals . (Hint : Consider th e leas t 7 suc h tha t fo r som e se t x o f ran k 7 , j(x) ^ x.) 1.1.8 Definition . Th e critical point cp(j) o f a nontrivia l embeddin g j i s the leas t ordina l 7 suc h tha t 7(7 ) ^ 7 . A cardina l ft i s strongly inaccessible i f ft is regula r an d 2 7 ft fo r eac h 7 ft. W e leav e i t a s a n exercis e t o sho w tha t measurabl e cardinal s ar e strongly inaccessible . I f ft i s a measurabl e cardina l an d \i i s a ft-complete ultrafilter o n ft, the n ft i s the critica l poin t o f the induce d embedding . LEMMA 1.1.9 . The following are equivalent for a cardinal ft. (1) ft is measurable. (2) There exists a nontrivial elementary embedding j : V M with critical point ft. P R O O F . Fo r th e forwar d direction , le t U b e a ft-complete nonprincipa l ultrafilter o n ft, an d le t j : V M b e th e induce d embedding . Fo r eac h set x , le t f x b e th e constan t functio n fro m ft t o {x}. T o se e tha t ft i s th e critical poin t o f j , firs t not e tha t fo r eac h a ft, f a represent s a 1 sinc e U i s ft-complete. No w le t i : ft ft be th e identit y function . The n fo r al l a ft, [/a]t/ [i]t / [f K ]u, S O j(ft) ft. For th e othe r direction , suppos e tha t j : V M i s a nontrivia l elemen - tary embeddin g wit h critica l poin t ft. Defin e ( 7 C V(K) b y lettin g E eU ^ E C KAKEJ(E). Then [ 7 i s a n ultrafilte r o n ft, an d sinc e th e critica l poin t o f j i s ft, i f E a (a 7) ar e member s o f [7 , for som e 7 ft, the n j(f|£ Q )= f]j(E a ), which ha s ft as a member , s o U is closed unde r intersection s o f size les s tha n ft. Furthermore , sinc e ft is not i n the imag e o f V unde r j , U is nonprincipal , so U witnesse s tha t ft i s measurable . More generally , i f j : V —• M i s a elementar y embeddin g wit h critica l point ft, a i s a se t i n M an d X i n V i s such tha t a G j(X), the n w e can use j t o defin e a n ultrafilte r U* o n X . Not e firs t o f al l tha t fo r eac h a G M there i s suc h a se t X i n V wit h a E j(X). Fo r instance , w e ca n choos e a such tha t a rank(a). The n X = V a work s sinc e J'(^a) = (^'(a)) M = Vj(a) fl M D F Q H M. Define £/ ^ C V{X) b y lettin g 4 17 * if and onl y i f a G j(A). The n U* i s an ultrafilte r o n X, an d sinc e th e critica l poin t o f j i s «, [/ * i s closed unde r
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