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 ^ . w l 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
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