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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