Uniformization and the diversity of Whitehead groups

Paul C. Eklof; Alan H. Mekler; Saharon Shelah

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Uniformization and the diversity of Whitehead groups

Paul C. Eklof·Alan H. Mekler·Saharon Shelah

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Abstract

The connections between Whitehead groups and uniformization properties were investigated by the third author in [Sh:98]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, aleph_1-coseparable) group of cardinality aleph_1 if and only if there is a ladder system on a stationary subset of omega_1 which satisfies 2-uniformization (respectively, omega-uniformization). These techniques allowed also the proof of various independence and consistency results about Whitehead groups, for example that it is consistent that there is a non-free Whitehead group of cardinality aleph_1 but no non-free aleph_1-coseparable group. However, some natural questions remained open, among them the following two: (i) Is it consistent that the class of W-groups of cardinality aleph_1 is exactly the class of strongly aleph_1-free groups of cardinality aleph_1 ? (ii) If every strongly aleph_1-free group of cardinality aleph_1 is a W-group, are they also all aleph_1-coseparable? In this paper we use the techniques of uniformization to answer the first question in the negative and give a partial affirmative answer to the second question.

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Paul C. Eklof, Alan H. Mekler, Saharon Shelah. 1992-04-15. Uniformization and the diversity of Whitehead groups. https://arxiv.org/abs/math/9204219

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