12 1. ELEMENTAR Y EMBEDDING S
Since C G jo(^i), w e may fi x a function he '• Xo V suc h tha t jo(hc)(do) =
C,
z = { d G X0 | h c(d) G /xi} G /i0
and d o G jo(^) « Le t S = {/ic(d ) | d G 2:} . W e hav e tha t XQ G V^ , s o
|Xo| K\. Therefore , |*S | K\ an d s o T = f]S G /xi , sinc e /i i i s KI-
complete. Sinc e T n - A 7 ^ 0 , i t suffice s t o sho w tha t .70(a ) £ C fo r eac h
a G T . W e hav e essentiall y show n thi s already : th e constan t functio n f
a
from XQ t o {a } represent s .70(a ) i n th e /io-ultrapower , he represent s C , an d
{d G
XQ
I fa{d) G /ic(^)} contain s z an d s o i s i n
JJLQ.
Thi s prove s Clai m 3 .
Proof o f Clai m 1: W e ar e give n function s / an d g i n MQ from jo(Xi) t o
the ordinal s suc h tha t th e se t
C = {bejo(X
1
)\f{b)=g(b)}
is i n jo(fii). Fo r eac h a G Xll 7r(/)(a ) = f(jo(a)) an d 7r(p)(a ) = p(jo(a)) .
We hav e t o sho w tha t
{a G Xi I 7r(/)(a) = 7r( 5)(a)} G /ii.
Suppose t o th e contrar y tha t
A = {aeX l\ n(f)(a) ± Tr(g)(a)} e fii.
By Clai m 3 , ther e exist s a G A suc h tha t jo {a) £ C . Fixin g suc h a n a , w e
have tha t n(f)(a) 7 ^ n(g)(a)1 whic h mean s tha t f(jo(a)) ^ g(jo(o)), whic h
gives a contradictio n sinc e jo(a) G C. Thi s prove s Clai m 1, an d th e proo f o f
Claim 2 i s almos t identical .
We no w hav e tha t n induce s a n orde r preservin g 1-1ma p
7f : (Ord^Xl^)Mo/jo(fJLi) - Ord Xl/in.
We hav e t o sho w tha t n i s onto . Thi s amount s t o showin g tha t fo r eac h
function / : X\ Ord ther e exist s a functio n g: jo(Xi) Ord suc h tha t
g G MQ and n(g) = f mo d \i\. Th e difficult y i s that th e functio n / nee d no t
be i n Mo .
For eac h a G X\ , le t h a: Xo » Ord b e a functio n whic h represent s
/(a) , i.e. , a functio n suc h tha t jo(h a)(do) = /(a) , wher e d o a s befor e i s th e
element o f M o represente d b y th e identit y functio n o n XQ . Defin e a functio n
G: Xo Ord Xl b y lettin g G(b) b e a functio n fro m X\ t o Or d suc h tha t
G(b)(a) = h
a
(b)
for al l a G X\. Le t g = jo(G)(do) . Applyin g jo, w e se e tha t fo r eac h
d G jo(Xo), jo(G)(d) i s a functio n fro m jo(X\) t o th e ordinals , s o g i s suc h
a functio n an d g G Mo. Finally , w e sho w tha t 7r(g) f. Fi x a E X\. B y
the definitio n o f 7r ,
^(«7)(«) = 9(h(a)) = Uo(G)(d 0))(Jo(a)) = jo(h a)(d0) = f(a),
Previous Page Next Page