8 1. ELEMENTAR Y EMBEDDING S
PROOF O F THEORE M
1.1.5 . Assum e towards a contradiction tha t V =
L an d tha t K is th e leas t measurabl e cardina l i n L. The n ther e i s a n el -
ementary embeddin g j : V M induce d b y a ^-complet e ultrafilte r U o n
K. Sinc e V = L , M mus t b e L also . B y elementarity , J(K) i s th e leas t
measurable cardina l i n M , bu t thi s give s a contradiction, sinc e M = L , « is
the leas t measurabl e cardina l i n L , an d ft J(K). D
As a second proof, w e present a property o f measurable cardinal s whic h
will b e ke y t o man y argument s i n thi s book . Firs t w e not e a genera l fac t
about elementar y submodels .
LEMMA
1.1.18Let . M be a transitive model of ZFC - Replacement. Let
t G X be sets in M and let Z an elementary submodel of M with X G Z.
Assume that one of the following holds.
Ord C M,
M is a set and cof{M n Ord) \X\,
Z H Ord is cofinal in M H Ord.
Then Z[t] = {f(t)
| / : I - M A / G 2 }
is an elementary submodel of M.
PROOF.
W e sho w tha t Z[t] satisfie s th e Tarski-Vaugh t criterion . Fi x a
formula (j) an d x\, ..., x
n
G Z[t] an d suppos e tha t
M | = 3y0(y,xi,...,x
n
).
Let / i , . . . , f
n
b e function s i n Z wit h domai n X suc h tha t fi{t) X{ holds
for eac h i. I f Ord C M o r \X\ cof(M n Ord) , the n ther e i s a n 7 7 G M
such tha t fo r eac h s G X, i f
M h 3 # ( y , / i (
S
) , . . , / (
S
) ) ,
then ther e exist s a y G M o f rank les s than7 7 suc h tha t
Mh0(y,/i(«),---,/n(s)) ,
and s o there i s such a n rj in Z. Likewise , i f Z H Ord i s cofinal i n M H Ord
then ther e exis t a n 7 7 G Z an d a y G M o f rank les s than7 7 suc h tha t
In eithe r case , there exist s i n Z a wellordering o f the set s i n M o f rank les s
than7 7 an d s o by Comprehensio n ther e i s a functio n g : X M i n Z suc h
that fo r eac h 5 G X1 i f there exist s a y o f rank les s than7 7 suc h tha t
then
MN^sWJi W /»(«))
The existence of such a g suffices t o show that Z[t] \= 3y0(y , #i,..., xn). D
1.1.19 Remark . I n ou r application s o f Lemm a 1.1.18th , e structure s M
will typicall y b e set s o f th e for m V
1
. Anothe r versio n o f th e lemm a hold s
for set s o f the for m #(7 ) fo r regula r 7 , withou t th e assumptio n tha t H(j)
satisfies th e Powerse t Axiom .
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