1.1. ULTRAPOWER S
3
principal an d nonprincipal fo r measure s i n th e sam e wa y tha t w e us e the m
for ultrafilters , an d us e th e term s measure an d ultrafilter interchangeably .
The existenc e of a countably complet e nonprincipa l measur e i s equivalen t
to th e existenc e o f a measurabl e cardinal .
LEMM A
1.1.4 . Suppose that U is a countably complete nonprincipal ultra-
filter on a set X
7
and let K be the completeness ofU. Then K is measurable.
P R O O F .
Le t A
a
(a
K)
b e set s in U such that H{A
a
: a
K}
0 U. Sinc e
the completenes s K of U i s infinite , b y addin g on e mor e se t i f necessar y w e
may assum e tha t 0{A
a
: a K} = 0 . Sinc e U i s close d unde r intersection s
of siz e /c , by replacin g eac h A
a
wit h
K=n
A
0
,
(3a
and the n thinnin g th e sequenc e i f necessary , w e ma y assum e tha t fo r eac h
nonzero a n, A
a
i s a prope r subse t o f n{A ^ : (3 a}. Le t Bo = X \ Ao ,
and fo r eac h nonzer o a n le t
B*= f]A
p
\Aa.
Then th e B
a
}s
ar e pairwis e disjoint , eac h B
a
i s th e complemen t o f a se t i n
[/, an d th e unio n o f th e B
a
's i s X. Defin e a n ultrafilte r U* o n n b y lettin g
each E C K be i n U* i f an d onl y i f U{B
a
: a G E} G U. The n sinc e U i s a
^-complete ultrafilter , [7 * must b e on e also .
Theorems 1.1. 5 an d 1.1. 6 sho w that tw o of the earlies t consistenc y result s
produced b y th e metho d o f forcin g ca n als o b e don e withou t forcing , fro m
measurable cardinals . Th e first o f thes e wa s know n befor e th e discover y o f
forcing, bu t th e secon d wa s discovere d afterwards .
T H E O R E M
1.1.5(134]^) . If there is a measurable cardinal then V ^ L.
T H E O R E M
1.1.6([24]) . Assume that there are uncountably many mea-
surable cardinals. Then the Axiom of Choice fails in L{Ord
u).
The mode l L(Ord
u)
ha s severa l equivalen t definitions , th e easies t o f
which i s probabl y
L{Ord")= | J L(V Ul(a)).
a^Ord
This model , first studie d i n [6] , i s know n a s th e Chan g model , an d i s th e
smallest inne r mode l o f Z F containin g ever y countabl e sequenc e o f ordinal s
in V .
Before provin g Theorem s 1.1. 5 an d 1.1.6w , e revie w som e basi c fact s
about ultrapower s an d measurabl e cardinals .
We sa y tha t a n embeddin g i s nontrivial i f i t i s no t th e identit y ma p o n
its domain .
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