the axioms of incidence

Before you can talk about distance, angle, or which point sits between two others, you need the most basic facts of all: which points lie on which lines, and which lie in which planes. That bare relationship — a point 'lies on' a line, a line 'lies in' a plane — is called incidence, and the axioms of incidence are the first family in Hilbert's system. They say nothing about size or order; they only pin down what is joined to what.

For the plane there are three of them. (I-1) Through any two distinct points there is exactly one line — two dots determine a unique line. (I-2) Every line contains at least two points — a line is not empty or a single dot. (I-3) There exist at least three points that do not all lie on one line — the plane is genuinely two-dimensional, not collapsed onto a single line. In space, further incidence axioms govern planes (three non-collinear points determine exactly one plane, and so on). From these alone, with no notion of measurement, you can already deduce real theorems: for instance, two distinct lines meet in at most one point, because if they shared two points, axiom I-1 would force them to be the same line.

Incidence geometry is the thin skeleton on which everything heavier is later hung. It also stands on its own: tiny finite 'geometries' satisfy the incidence axioms — the smallest has just three points and three lines, like the corners and sides of a triangle — and studying them shows you how much (and how little) the incidence axioms by themselves can decide. Projective and affine geometries are built by varying exactly this layer.