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 -
saturated ideal, then 2 = K . More generally, if X K and if K carries
a normal X-saturated ideal then 2 \.
Theorem 3.3.1. Assume thatfcfc.carries anfc*-saturated ideal. Then
a) If 2 u » ^ then 2 « ^.
*0 * 1 *0
b) If Nj. 2 Nw then 2 = 2 •
c) If 2 = tf then 2 ^ .
Theorem 3.2.1. Assume that H. carries a X-saturated ideal and that
2 •» . Then
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
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
Theorem 4.2.4. Assume that fcL carries anfcL-saturatedideal. If tf is
± z a)
a strong limit cardinal then