6 1. ELEMENTAR Y EMBEDDING S
For a give n filte r F o n a se t X , w e sa y tha t a se t A C X i s F-positive i f
A intersect s eac h membe r o f F. I f I i s a n idea l o n a se t X , I
+
denote s th e
set o f F-positiv e subset s o f X , wher e F i s th e filte r dua l t o I. Equivalently ,
I+=V{X)\I.
1.1.11 Definition . A filte r F o n a n ordina l K, is normal i f i t i s close d
under diagona l intersections , i.e. , i f fo r al l (A
a
: a K) C F , th e diagona l
intersection
A{Aa :aK) = {(3K\[3e f] A a}
a(3
is als o i n F.
Equivalently, F i s norma l i f eac h regressiv e functio n whos e domai n i s
F-positive i s constan t o n a n F-positiv e set . A n idea l i s sai d t o b e normal i f
its correspondin g filte r i s normal .
LEMMA 1.1.12 . Let j : V * M be a nontrivial elementary embedding
with critical point K and let U {^ 4 C K \ K G j(A)}. Then U is a
nonprincipal normal ultrafilter on K.
P R O O F .
Th e proo f o f Lemma 1.1. 9 show s tha t U i s a nonprincipa l ultra -
filter. T o show that U i s normal, fi x / : n n suc h tha t {a | f{a) a} G U.
Then
* £
j(ia
I
f(
a)
&}),
s o
J(f)(
K)
K- Le t (3 = j(f)(K) an d le t A = {a K \ f(a) = /?} .
Then j{A) = {a J(K) | j(f)(a) = j(f3)}. Sinc e j(0) = 0, K G j(A), s o
AeU.

It follow s immediatel y fro m th e definitio n o f normalit y tha t i f U i s a
nonprincipal norma l ultrafilte r o n a n uncountabl e cardina l K then U i s K-
complete, an d s o i n particular , b y Lemm a 1.1.10V , K/U i s wellfounded .
LEMMA 1.1.13 . If U is a nonprincipal normal ultrafilter on K and i is
the identity function on n
7
then [i}u K.
P R O O F .
Fo r eac h a ft, le t f
a
b e th e constan t functio n fro m
K
t o {a}.
Then fo r eac h pai r a (3 below K,
{7 « I fad) 7/3(7) } = « e U,
and fo r eac h a K,
{7 * I fM i(7) } = « \ ( a + 1) G tf.
This show s tha t [i]f / K. NO W i f # i s a functio n wit h domai n n an d
{7
A C
I 5(7) i(7) } G U,
then b y th e normalit y o f U ther e i s a n a K such tha t
{7 « I 5(7) = /a(7) } ^
which show s tha t [zj^ y = ft.
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