In this paper we present a systematic study of ideals over uncountable
sets. We are particularly interested in the role that various properties of
ideals play in the investigations of arithmetic of cardinal numbers. We show
that there is a deep relationship between saturation of ideals and the cardinal
function 2 ; we study consequences of existence of a certain kind of ideals
for the generalized continuum hypothesis and the singular cardinals problem.
Our applications of ideals over uncountable sets use two methods which
are just two different ways of expressing the same phenomena: the method of
generic ultrapowers and the method of almost disjoint functions.
The method of generic ultrapowers is a combination of Cohen's method of
forcing with the method of ultraproducts used in model theory and in the theory
of large cardinals. Given an ideal I over an uncountable set, we use the
sets which are not in I as forcing conditions, and in the generic extension
of the universe so obtained, we construct the ultrapower of the ground model,
using the generic ultrafilter. Then we combine the technique of ultraproducts
with the method of forcing to obtain useful information about the ground model.
An argument of this kind was first used by Solovay in [15] where he used
an ultrapower in the Boolean-valued model to derive some properties of real
valued measurable cardinals. In his contribution to the singular cardinals
problem [14] Silver used a non-well-founded ultrapower in a generic extension
to show that if K is a singular cardinal of uncountable cofinality and if the
K +
G.C.H. holds below K then 2 = K . Bot these arguments can be considered a
special case of a general method of generic ultrapowers.
* Received by the editors September 12, 1977
Research supported by NSF grant MSP 76-05993
Research supported by NSF grant GP-43841
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