Absolutely Rigid Systems and Absolutely Indecomposable Groups

We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal κ(ω). Added December 2004: The proofs of Theorems 0.2 and 0.3 are not correct, and the claimed results remain open. (The ”only if” part assertion in the last 3 lines before ”Proof of (II)” on p. 266 is not correct.) However, Theorems 0.1 and 0.4, which give upper bounds to the size of rigid systems/groups, are valid. And the construction in the proof of Theorem 0.3 does yield an affirmative answer to Nadel’s question whether there is a proper class of torsion-free abelian groups which are pairwise absolutely non-isomorphic.