14 THOMAS JECH AND KAREL PRIKRY

there is an unbounded I-function g with dom(g) c_ dom(f) such that

g(a) f(a) for all a € dom(g). We construct a sequence F F ...

F ... of functionals on S as follows: Let F = {d} where d

n 0

is the diagonal function on S : d(a) = a for all a € S. Given F , let

n

us consider for each f € F a family H of unbounded functions g with

dom(g) £ dom(f) and g(a) f(a) for all a, such that H_ is maximal with

the property that dom(g ) ( 1 dom(g ) € I whenever g and g are distinct

members of H,,. Then let F ,, = U{H^ : f € F }. •

f n+ 1 f n

Theorem 1.4.3. The ideal I = {X £ K : |x| K} is not precipitous.

Proof. A set X £ K has positive measure just in case |x| - K. For each

such X, let f be the unique order preserving function from X onto K.

X

For each set X of positive measure there exists a set Y c X of

positive measure such that fv(ct) fv(a) for all a € Y : let Y = » {a 6 X :

Y X

f (a) is a successor ordinal}, and f (a) = fv(a)+l for all a € Y. Thus

X X A

for every set X of positive measure there is an I-partition W of X such

that for all Y € W , we have fv(a) fv(a) for all a € Y.

X x X

We shall construct a descending sequence of functionals on K. First we

construct I-partitions Wrt W ... W ... of K as follows: We let

0 1 n

W~ = {K}, and for each n, we let W , = U{WV : X € W }. Then for each n,

U n+l X n

we let F be the functional F = {f : X £ W }. It is clear that F„ F.

n n X n 0 1

... F ... and hence I is not precipitous, n

We conclude this section with the remark that existence of a precipitous

ideal is equiconsistent with existence of a measurable cardinal; more pre-

cisely:

Theorem 1.4.4 (see [7]). a) If a regular uncountable cardinal K carries

a precipitous ideal then K is measurable in some transitive model of ZFC.

b) If K is a measurable cardinal then there exists a generic model in