Geometry of Definable Sets

In the previous chapter we saw examples of mathematical structures that are simple enough to allow a complete analysis of the parametrically definable subsets of their domains. Those structures are of interest, but the real objects of study in mathematics are richer structures such as \(({\mathbb {Q}},+,\cdot )\) or \(({\mathbb {R}},+,\cdot )\) . To talk about them we first need to take a closer look into their definable sets. Definable sets in each structure form a geometry in which the operations on sets are unions, intersections, complements, Cartesian products, and projections from higher to lower dimensions. We will see how those operations correspond in a natural way to Boolean connectives and quantifiers, and how the name “geometry” is justified when it is applied to sets definable in the field of real numbers. The last two sections are devoted to a discussion of the negative solution to Hilbert’s 10th problem.