A P-adic structure which does not interpret an infinite field but whose Shelah completion does

Abstract

We give a $p$-adic example of a structure whose Shelah completion interprets $\mathbb{Q}_p$ but which does not (provided an extremely plausible conjecture holds) interpret an infinite field. In the final section we discuss the significance of such examples for a possible future geometric theory of $\mathrm{NIP}$ structures.