Growing Spines: Ad Infinitum et Ad Infinitesimalia

Abstract

We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$ such that $G\preccurlyeq G\oplus H$ for some non-trivial $H$. We apply this to characterize which ordered abelian groups (respectively fields) ensure that any henselian valuation with said value group (respectively residue field) is definable in the language of rings. This answers a question of Krapp, Kuhlmann, and Link.