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 }. 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
Previous Page Next Page