10 1. ELEMENTAR Y EMBEDDING S
1.1.22 R e m a r k . Th e argumen t i n th e paragrap h abov e show s tha t L sat -
isfies th e followin g statement , whic h i s i n fac t consisten t wit h man y larg e
cardinals: Ther e exist s a functio n h: uo\ uo\ such tha t fo r an y countabl e
elementary submode l X o f 7
7
, wher e 7 i s th e first stron g limi t cardinal ,
o.t.(XnOrd) / i ( l n w i ) .
We now turn ou r attentio n t o th e proo f o f Theorem 1.1.6 . Unde r Chang' s
original definitio n [6] , the mode l L(Ord
u)
(ther e calle d C^
1
) i s th e resul t o f
relativizing th e constructio n o f L t o th e languag e C
{JJ1UJ11
whic h allow s fo r
countably infinit e block s o f quantifier s an d countabl y infinit e conjunction s
and disjunction s o f formulas . Th e Chan g mode l i s the n define d b y startin g
with CQ 1 = 0 , lettin g C^
+1
b e al l th e subset s o f C^ 1 definabl e ove r C^ 1 b y
formulas i n C
U1UJ1
usin g a countabl e se t o f parameter s fro m C^ 1 , an d takin g
unions a t limi t stages . Equivalently , Th e Chan g mode l ca n b e define d a s
the increasin g unio n o f a clas s o f set s N a, a G Ord, wher e NQ = 0 , union s
are take n a t limi t stages , an d eac h N a+i i s th e collectio n o f subset s o f N
a
definable ove r N
a
(i n ou r usua l language ) wit h an y countabl e sequenc e fro m
a a s a parameter. W e leav e it t o th e reade r t o chec k the followin g ke y points .
1.1.23 Exercise . Th e thre e definition s o f th e Chan g mode l w e hav e give n
define th e sam e class , an d thi s clas s satisfie s ZF . (Hint : Th e onl y nontrivia l
case i s Replacement, an d thi s follow s fro m th e definabilit y o f L(Ord
u)
(bot h
inside th e mode l an d i n V) an d th e fac t tha t Replacemen t hold s i n V.)
1.1.24 Exercise . Th e Chan g mode l satisfie s it s definitio n insid e itself , an d
every elemen t o f th e Chan g mode l i s S 2 definabl e insid e th e mode l fro m a
countable sequenc e o f ordinals .
Before provin g Theore m 1.1. 6 w e presen t tw o technica l lemmas .
LEMM A
1.1.25 . Suppose that ^o?M i are countably complete nonprincipal
ultrafilters on sets Xo and X\, respectively. For each i G {0,1}7 let K{ be
the completeness of Hi, and let ji\ V Mi be the associated elementary
embedding. Assume that Xo G VKl. Let JQ be the ultrapower map computed
in Mi using ji(/io )
an
d let be the ultrapower map computed in MQ using
jo {ill). Then j% \Ord = j
0
\Ord and j% \Ord = ji \Ord.
P R O O F .
Sinc e X
0
G V K1,
^0 = Cp(jo) Cp(ji) = Ki.
We sho w first tha t jl\Ord = jo\Ord. Not e tha t /i
0
C V(Xo), s o /i
0
G V
Kl
,
and thu s ji(/mo) Mo- Sinc e Mf 1 c Mi , M*° c Mi , s o th e clas s Ord
when compute d i n M i i s th e sam e a s whe n compute d i n V. Sinc e th e
restriction t o th e ordinal s o f th e ma p derive d fro m /x o depend s onl y o n
Ordx°
an d /ZQ , w e hav e tha t jo\Ord = j$ \Ord.
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