Ideals over Uncountable Sets: Application of Almost Disjoint Functions and Generic Ultrapowers: Application of Almost Disjoint Functions and Generic Ultrapowers
This work is a systematic study of ideals over uncountable sets. In particular, we investigate the role of various properties of ideals in arithmetic of cardinal numbers. We also study consequences of existence of precipitous ideals for the generalized continuum hypothesis and the singular cardinals problem.
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81 carries 82-saturated aleph function Assume that 81 assumption Boolean-valued model branching family cardinal in M[G cardinal number cardinal of cofinality carries a precipitous closed unbounded set complete Boolean algebra countable disjoint family disjoint functions disjoint sets elementary embedding exists a family f and g fixed point following theorem function f G C F Galvin and Hajnal hence I-function f I-generic M-ultrafilter I-partition implies L-function least value least weakly inaccessible Lemma Let F let G let us assume measurable cardinal model of ZFC nice cardinal function nonprincipal M-K-complete M-ultrafilter nonregular ultrafilter notion of forcing partial functions positive measure precipitous ideal precipitous X-saturated ideal regular cardinal regular uncountable cardinal saturated set of positive singular cardinals problem stationary subset strong limit cardinal successor cardinal Theorem transitive model ultrapower ultraproducts Ultu Ultu(M uncountable sets uniform ultrafilter weakly inaccessible cardinal weakly normal Xth value