1.1. ULTRAPOWER S 13
by th e choic e o f h
a
. Thi s verifie s tha t n induce s a n orde r preservin g bijectio n
7f : Ordj°(Xl)/jo(in) - Ord Xl/^
Putting everythin g together , w e ge t finally tha t j \ \Ord = \Ord. T o
see this , not e tha t fo r eac h ordina l 7 , i f g i s th e constan t functio n fro m
X\ t o {7} , the n [g]
/xi
= ji(7) . Likewise , i f / i s th e constan t functio n fro m
jo(^i ) t o {7} , then [/]
Jo(m
) = j?(7) - Bu t ix(g) = / , an d sinc e TT induce s th e
isomorphism 7f , thes e tw o value s ar e th e same .
1.1.26 Remark . Lemm a 1.1.2 5 ca n fai l i f w e dro p th e requiremen t tha t
Xo E V
Kl
, i.e. , tha t |Xo | i s les s tha n th e completenes s o f /xi .
LEMM A
1.1.27Given . an ordinal 7 , let A
1
be the class of
K
such that
there exists a countably complete nonprincipal ultrafilter \i of completeness
K such that j (7) ^ 7 , for j : V " M t/i e embedding induced by fi. Then for
all ordinals 7 , ^4
7
i s finite.
P R O O F .
Suppos e tha t th e lemm a fails , an d le t 7 b e th e leas t ordina l p
such tha t A
p
i s infinite . Fi x th e followin g object s
(fii : i CJ) , a sequenc e o f countabl y complet e nonprincipa l ultra -
filters,
a n increasin g sequenc e o f cardinal s (KI : i u) suc h tha t eac h Ki is
the completenes s o f \i{
fo r eac h i a; , th e elementar y embedddin g ji : V Mi give n b y
Mi,
and suppos e tha t fo r eac h i u, ^(7 ) 7 ^ 7.
By Lemm a 1.1.25 , fo r eac h intege r k 0 , j£[Or i = j^fOrd , wher e j £ i s
the embeddin g compute d i n MQ from jo(Mfc) - S o
M
0
| = V f c a ; \ { 0 } j g ( 7 ) ^ 7 -
In Mo , Jo(7) i s the leas t p such tha t A
p
i s infinite, whic h mean s tha t jo(7 )
7. Sinc e jo(7) canno t b e les s tha n 7 , jo(7 ) = 7 which give s a contradiction .

PROOF
O F T H E O R E M
1.1.6 . Le t (/c
a
: a
UJ{)
b e th e first a;i-man y
measurable cardinals . Fo r eac h a , le t \i
a
b e a tt
a
-complete nonprincipa l
ultrafilter o n AC
Q
, and le t j
Q
: V M
a
b e th e induce d embedding . Le t A be
the supremu m o f th e «
a
's. W e hav e tha t A w C L(Ord u). Suppos e toward s
a contradictio n tha t ther e i s a wellorderin g n o f A ^ i n LiOrd?). The n TT i s
^-definable i n L{Ord
u)
usin g a countabl e se t s o f ordinal s a s a parameter .
Fix a regular cardina l 7 max{A , ran/c(7r), sup(s)} an d not e tha t j
a
( 7 ) = 7
for al l a . B y makin g 7 sufficientl y large , an d possibl y addin g t o s , w e ca n
assume tha t TT i s definabl e i n l/
7
n LiOrd^) fro m 5 . I n particular , w e ma y
fix a formul a (j) suc h tha t
TT = {
a
E A
w
x A ^ I V1 n L{Ord") | = 0(a , s)}.
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