Whitehead's problem and reverse mathematics
type: Item
instance of: scholarly article
title: Whitehead's problem and reverse mathematics
zbMATH Open document ID: 1405.03033
author: Yang Sen
published in: Israel Journal of Mathematics
publication date: 2018-11-13T00:00:00Z
review text: 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.
reviewed by: Jeffry L. Hirst
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03D80
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 6976741
zbMATH Keywords: reverse mathematics
zbMATH Keywords: computable algebra
zbMATH Keywords: constructive algebra
zbMATH Keywords: Whitehead group
zbMATH Keywords: free group
zbMATH Keywords: Prüfer
zbMATH Keywords: Pontryagin
MaRDI profile type: Publication
full work available at URL: s11856-018-1770-5
OpenAlex ID: W2886194079
cites work: Algorithmic Randomness and Complexity
cites work: Effectively categorical abelian groups
cites work: Whitehead's Problem is Undecidable
cites work: Q5609457
cites work: Independence in computable algebra
cites work: Set Theory
cites work: ∏ 0 1 Classes and Degrees of Theories
cites work: On a problem of Baer and a problem of Whitehead in abelian groups
cites work: Infinite Abelian groups, Whitehead problem and some constructions
cites work: Q3395521
cites work: Reverse mathematics and fully ordered groups
cites work: Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem
DOI: 10.1007/S11856-018-1770-5
Recommended article: Set Theory Generated by Abelian Group Theory
Recommended article: Q3949041
Recommended article: Algebraic disguises of \(\Sigma ^ 0_ 1\) induction
Recommended article: Q4267664
Recommended article: Q3464625
Recommended article: Factorization of polynomials and \(\Sigma ^ 0_ 1\) induction
Recommended article: Q5711879
Recommended article: Minimal prime ideals and arithmetic comprehension
Recommended article: Q3481699
Recommended article: Countable algebra and set existence axioms
description: scientific article; zbMATH DE number 6976741