Definable coordinate geometries over fields, part 2: applications

Abstract

In Part 1 of this study we showed, for a wide range of geometries, that the relationships between their concept-sets are fully determined by those between their (affine) automorphism groups. In this (self-contained) part, we show how this result can be applied to quickly determine relationships and differences between various geometries and spacetimes, including ordered affine, Euclidean, Galilean, Newtonian, Late Classical, Relativistic and Minkowski spacetimes (we first define these spacetimes and geometries using a Tarskian first-order language centred on the ternary relation $\mathsf{Bw}$ of betweenness). We conclude with a selection of open problems related to the existence of certain intermediate geometries.