4 1. ELEMENTAR Y EMBEDDING S
1.1.7 Exercise . Sho w tha t i f M an d N ar e transitiv e model s o f ZF C an d
j : M N i s a nontrivia l So-elementar y embeddin g (i.e. , preservin g th e
^-relation) wit h N C M , the n j i s no t th e identit y o n th e ordinals . (Hint :
Consider th e leas t 7 suc h tha t fo r som e se t x o f ran k 7 , j(x) ^ x.)
1.1.8 Definition . Th e critical point cp(j) o f a nontrivia l embeddin g j i s
the leas t ordina l 7 suc h tha t 7(7 ) ^ 7 .
A cardina l ft i s strongly inaccessible i f ft is regula r an d 2 7 ft fo r eac h
7 ft. W e leav e i t a s a n exercis e t o sho w tha t measurabl e cardinal s ar e
strongly inaccessible . I f ft i s a measurabl e cardina l an d \i i s a ft-complete
ultrafilter o n ft, the n ft i s the critica l poin t o f the induce d embedding .
LEMMA 1.1.9 . The following are equivalent for a cardinal ft.
(1) ft is measurable.
(2) There exists a nontrivial elementary embedding j : V M with
critical point ft.
P R O O F .
Fo r th e forwar d direction , le t U b e a ft-complete nonprincipa l
ultrafilter o n ft, an d le t j : V M b e th e induce d embedding . Fo r eac h
set x , le t f
x
b e th e constan t functio n fro m ft t o {x}. T o se e tha t ft i s th e
critical poin t o f j , firs t not e tha t fo r eac h a ft, f
a
represent s a
1
sinc e U i s
ft-complete. No w le t i : ft ft be th e identit y function . The n fo r al l a ft,
[/a]t/ [i]t / [f
K
]u, S O j(ft) ft.
For th e othe r direction , suppos e tha t j : V M i s a nontrivia l elemen -
tary embeddin g wit h critica l poin t ft. Defin e ( 7 C V(K) b y lettin g
E eU ^ E C KAKEJ(E).
Then [ 7 i s a n ultrafilte r o n ft, an d sinc e th e critica l poin t o f j i s ft, i f E
a
(a 7) ar e member s o f [7 , for som e 7 ft, the n
j(f|£
Q
)= f]j(E a),
which ha s ft as a member , s o U is closed unde r intersection s o f size les s tha n
ft. Furthermore , sinc e ft is not i n the imag e o f V unde r j , U is nonprincipal ,
so U witnesse s tha t ft i s measurable .
More generally , i f j : V —• M i s a elementar y embeddin g wit h critica l
point ft, a i s a se t i n M an d X i n V i s such tha t a G j(X), the n w e can use
j t o defin e a n ultrafilte r U* o n X . Not e firs t o f al l tha t fo r eac h a G M
there i s suc h a se t X i n V wit h a E j(X). Fo r instance , w e ca n choos e a
such tha t a rank(a). The n X = V
a
work s sinc e
J'(^a) = (^'(a)) M = Vj(a) fl M D F
Q
H M.
Define £/ ^ C V{X) b y lettin g 4 17* if and onl y i f a G j(A). The n U* i s
an ultrafilte r o n X, an d sinc e th e critica l poin t o f j i s «, [/ * i s closed unde r
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