1.1. ULTRAPOWER S 5 intersections o f siz e les s than K. Further , U* i s nonprincipal i f an d onl y if there i s no b G V suc h tha t j(b) a. Continuing wit h j , a and X , defin e Y = {j(f)(a)\f:X^VAfeV}. Using th e Axio m o f Choic e an d th e Replacemen t scheme , i t follow s tha t Y - M. Le t M a b e th e transitiv e collaps e o f F, an d le t ka: M a -+ M be th e invers e o f th e collapsin g map . Thi s give s th e followin g commutin g diagram. V- 3 - - M There i s a natural isomorphis m n a : V x /U* 1", defined b y lettin g Note tha t thi s i s well-defined : i f w e hav e / an d g i n V x , the n the y ar e equivalent mo d U* i f and onl y if C = { c e I | / ( c ) = S ( c ) } G [ / a X , which hold s i f an d onl y i f a G j(C), whic h hold s i f an d onl y i f j(f)(a) = j(g)(a). The n j a : V M a i s the ultrapower ma p induced by U* (s o in fac t this i s independent o f the choic e of X, a s long a s a G j(X)). I t ca n happe n that Y M, i n whic h cas e M a i s als o equa l t o M an d k a i s th e identity . In particular , thi s happen s whe n a is the membe r o f M represente d b y th e identity functio n o n X. LEMMA 1.1.10 . Suppose that U is a n-complete ultrafilter on a set X, and let j : V —• M be the associated embedding. Then M K C M. PROOF. Le t i: X —* X be th e identit y functio n o n X , an d le t a [i\u. Then, usin g the terminology above , M a = M an d k a i s the identit y functio n on M , i.e. , M = {j(f)(a)\f:X^VAfeV}. Since U is ^-complete, the critical point o f j i s at leas t n. Le t (b a : a K) be a /^-sequenc e o f element s o f M. Fo r eac h a K, fix a functio n f a : X V such tha t b a = j(f a )(a)- Le t s (fa : a K). The n j(s) i s a sequenc e (fa :a 3 (*)) a n d (/* :a K) = (j(f a ) :aK). This sequenc e i s in M , s o (j(fa)(a) : a K) = (b a :a K) is also in M .
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