The Difficulty of Classifying Decomposable Torsion-Free Abelian Groups

An abelian group is decomposable if it can be written as a direct sum of two (or more) nontrivial subgroups. Otherwise it is indecomposable. The only indecomposable torsion groups are cyclic groups of the form $\mathbb{Z}(p^n)$, where $p$ is prime (as well as Pr\"ufer groups, $\mathbb{Z}(p^\infty)$). Mathematicians have been unable to describe the class of decomposable torsion-free groups. As it turns out, this problem is analytic complete (it cannot be characterized by a first-order formula). We show this by comparing it to the problem of determining whether an infinitely branching tree in $\omega^{<\omega}$ has an infinite path.