IDEALS OVER UNCOUNTABLE SETS 3

proved in section 3 are the following:

Theorem 3.1.2. Assume that 2 = a for all a K. If K carries a K -

K +

saturated ideal, then 2 = K . More generally, if X K and if K carries

K

a normal X-saturated ideal then 2 \.

Theorem 3.3.1. Assume thatfcfc.carries anfc*-saturated ideal. Then

*0 *L

a) If 2 u » ^ then 2 « ^.

*0 * 1 *0

b) If Nj. 2 Nw then 2 = 2 •

^0 *1

c) If 2 = tf then 2 ^ .

wl 2

Theorem 3.2.1. Assume that H. carries a X-saturated ideal and that

2 •» . Then

K

In section 4 we investigate the size of 2 where K is a strong limit

cardinal of uncountable cofinality. For clarity of exposition we restrict our-

selves to the typical case when cf K « a). We look for a bound on the size

K

of 2 and show that it is necessary to assume that K admits a description

in terms of smaller ordinals. We consider the notion of a nice cardinal func-

tion $ (this notion is introduced in section 3) and investigate the case

K = $(o)-|). (For instance, $(a) can be N , or $(a) can be the a car-

dinal with the property that a = tf .) Among others, we prove the following

theorems.

Theorem 4.2.4. Assume that fcL carries anfcL-saturatedideal. If tf is

± z a)

a strong limit cardinal then

\

2

v