This paper is devoted to o-minimalism, the study of the first-order properties of o-minimal structures. The main protagonists are the pseudo-o-minimal structures, that is to say, the models of the theory of all o-minimal L-structures, but we start with a more in-depth analysis of the well-known fragment DCTC (Definable Completeness/Type Completeness), and show how it already admits many of the properties of o-minimal structures: dimension theory, monotonicity, Hardy structures, and quasi-cell decomposition, provided one replaces finiteness by discreteness in all of these. Failure of cell decomposition leads to the related notion of a eukaryote structure, and we give a criterium for a pseudo-o-minimal structure to be eukaryote.
To any pseudo-o-minimal structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a nontrivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics. As an application, we study certain analytic subsets, called Taylor sets.