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We prove that it is consistent that there is a non-reflexive Whitehead group, in fact one whose dual group is free. We also prove that it is consistent that there is a group A such that Ext(A,Z) is torsion and Hom(A,Z)=0. As an application we show the consistency of the existence of new co-Moore spaces.arXiv:math/9908157v1 fatcat:ztzehpziyrc5rfyyd7n7762wj4