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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