Uniformization and the Diversity of Whitehead Groups

The connections between Whitehead groups and uniformization properties were investigated by the third author in [9]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, א1-coseparable) group of cardinality א1 if and only if there is a ladder system on a stationary subset of ω1 which satisfies 2-uniformization (respectively, ω-uniformization). (See also [5, §XII.3]; definitions are reviewed below.) 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 א1 but no non-free א1-coseparable group (cf. [5, XII.3.18]). However, some natural questions remained open, among them the following two, which are stated as problems at the end of [5, p. 454].