On the existence of precovers

It is proved consistent with ZFC + GCH that for every Whitehead group A of infinite rank, there is a Whitehead group H_A such that Ext (H _A, A) ≠ 0. This is a strong generalization of the consistency of the existence of non-free Whitehead groups. A consequence is that it is undecidable in ZFC + GCH whether every ℤ-module has a ^⊥{ℤ}-precover. Moreover, for a large class of ℤ-modules N, it is proved consistent that a known sufficient condition for the existence of ^⊥{N}-precovers is not satisfied.