CHAPTER 1 Elementary embedding s 1.1. Ultrapower s An elementary embedding is a mapping j: M N , wher e M an d N ar e structures ove r th e sam e languag e £ , suc h tha t fo r ever y n-ar y (fo r som e integer n ) formul a (j) in C and all finite sequence s a i , . . . , an fro m M , M ( = 0(ai, •, an) ^ N \= /(j(ai),... ,j(a n )). Given a set X, a /i/ter on X i s a nonempty collection of subsets of X whic h is closed under finite intersections and supersets. A n ideal on X i s a nonempty collection of subsets of X close d under subset s and finite unions, so for every filter F ther e i s a correspondin g idea l / , compose d o f the complement s of the member s o f F , an d vic e versa . A n ultrafilter o n X i s a filter U on X such tha t fo r al l A C X, exactl y on e of A an d X \ A i s in U. Ultrafilter s can b e used t o generate elementar y embeddings . Give n a n ultrafilte r U on a se t X , th e clas s ultrapowe r o f V correspondin g t o U, denote d b y eithe r Vx jJJ o r Ult(V, [/), i s the structure whos e elements ar e equivalence classe s of functions wit h domai n X unde r th e equivalence relatio n f ~ g ^{xeX\ f(x) - g(x)} e U, such tha t fo r eac h n-ar y relatio n A i n £ , A([/i]jy,... , [f n ]u) hold s i n th e structure Ult(V, U) (wher e [f}u denote s th e equivalence clas s containin g / ) if and only if {xeX\A(fx(x),...,fn(x))}£U. A classical theorem due to Los says that i f the following conditio n holds , then th e embeddin g fro m V int o Ult(V, U) whic h map s eac h x G V t o the equivalenc e clas s o f th e constan t functio n fro m X t o {x} i s elemen - tary: fo r eac h intege r n , fo r eac h (n + l)-ar y formul a j an d eac h sequenc e / i , . . . , f n o f functions wit h domai n X, there exists a function g in such tha t for eac h x G X, i f ther e exist s a z suc h tha t 4(z, /i(x),..., fn(x)) holds , then (f(g(x),fi(x),.. ., f n (%)) holds . Th e proo f follow s fro m a n inductio n on th e quantifier complexit y o f formulas, usin g th e conditio n abov e t o add existential quantifiers . W e call thi s embeddin g th e embeddin g induced by or associated to U. I f Ult(V, U) is wellfounded, the n w e identify i t wit h it s transitive collapse . W e adopt th e convention tha t th e notation j : V M , for an y embedding j , indicate s that M i s transitive. Whe n the image mode l may b e illfounded , w e writ e j : V (M, E), wher e E i s th e induce d G - relation. l http://dx.doi.org/10.1090/ulect/032/01
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