Reverse Mathematics and Fully Ordered Groups

Abstract

We study theorems of ordered groups from the perspective of reverse mathematics. We show that $\mathit{RCA}_0$ suffices to prove Hölder's Theorem and give equivalences of both $\mathit{WKL}_0$ (the orderability of torsion free nilpotent groups and direct products, the classical semigroup conditions for orderability) and $\mathit{ACA}_0$ (the existence of induced partial orders in quotient groups, the existence of the center, and the existence of the strong divisible closure).