Whitehead's problem and reverse mathematics
An abelian group \(G\) is Whitehead if for every abelian group \(P\) and every surjective homomorphism \(\pi : P \to G\) with \(\operatorname{ker}(\pi ) \cong \mathbb Z\), there is a homomorphism \(\rho : G \to P\) such that for all \(x\in G\), \(\pi (\rho (x)) = x\). Every free abelian group is Whitehead. The status of the converse is known as the Whitehead problem, and was shown to be independent of \(\text{ZFC}\) by \textit{S. Shelah} [Isr. J. Math. 18, 243--256 (1974; Zbl 0318.02053)]. The restricted converse, that every countable abelian Whitehead group is a free abelian group, was proved by \textit{K. Stein} [Math. Ann. 123, 201--222 (1951; Zbl 0042.08703)]. This paper analyzes Stein's theorem in the framework of reverse mathematics. The author shows that over \(\text{WKL}_0\), Stein's theorem is equivalent to \(\text{ACA}_0\). Additionally, Stein's theorem is shown to hold in the minimal \(\omega\)-model of \(\text{RCA}_0\) consisting of the computable sets. Another model of \(\text{RCA}_0 + \neg \text{WKL}_0\) in which Stein's theorem fails is also presented. Thus, over \(\text{RCA}_0\), Stein's theorem lies strictly between the implication \(\text{WKL}_0 \to \text{ACA}_0\) and the subsystem \(\text{ACA}_0\). Additional results on Prüfer bases and the Pontryagin property are proved in \(\text{RCA}_0\), and may be of use to computable and constructive algebraists.
scientific article; zbMATH DE number 6976741
scientific article