The decomposability problem for torsion-free abelian groups is analytic complete

2013/11/08 by Riggs, Kyle
#03C57 #03D45 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1311.1865

Abstract

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is Σ⁰₃-complete. However, when we consider groups of infinite rank, it becomes Σ¹₁-complete, so it cannot be characterized by a first-order formula in the language of arithmetic.

Related

• Decompositions of torsion-free abelian groups
• Completely decomposable direct summands of torsion--free abelian groups\n of finite rank
• Canonical Decompositions of Abelian Groups
• Decomposability of Finitely Generated Torsion-free Nilpotent Groups
• A proof of the Borel completeness of torsion free abelian groups