 In the axiomatic geometries that Hilbert put forward, the fundamental objects (points, lines, planes) are not defined. Instead, Hilbert specified the network of relations that links them, which determines how they can be used and what can be said about them. He presented five families of axioms, sorted according to the concepts they codified. He then created a variety of geometries obeying a variety of axioms systems, and established the consistency of them by giving them coordinates over suitable rings and fields—often his geometries admit many interpretations or models. Here Hilbert took advantage of his expertise in the novel theories of number fields and rings. This gave these geometries all the consistency of arithmetic, and increased Hilbert’s interest in attempts to ground arithmetic in some form of set theory and logic.