a model

Axioms are written with undefined words — point, line, between — that mean nothing on their own. A model is what happens when you assign concrete meanings to those words using objects you already understand, in such a way that every axiom comes out true. It is a faithful interpretation: a particular world in which the bare grammar of the axioms is realized as honest fact. The axioms describe a shape; a model is a thing of that shape.

To build one you do three things: pick a set to be the 'points', say what a 'line' is, define the relations like 'between' and 'congruent' — then check, one axiom at a time, that each becomes a true statement under your dictionary. The standard model of Euclidean geometry interprets 'point' as a pair of real numbers (x, y), 'line' as the set of solutions of a linear equation, distance by the Pythagorean formula, and so on; every Hilbert axiom is then a verifiable fact about real numbers. But the same axioms can have very different-looking models. The Poincare disk reinterprets 'point' as a point inside a circle and 'line' as a special arc, and it satisfies all the neutral axioms while breaking the parallel one — a wholly different world obeying almost the same rules.

Models are the workhorses of foundations because they convert hard logical questions into the existence of structures. Want to prove an axiom system is consistent? Produce one model — a structure cannot contain a real contradiction. Want to prove an axiom is independent? Produce a model where the others hold and it fails. Want to know whether the axioms pin down a unique world or leave room for variety? Compare their models: if all models are essentially the same, the system is categorical. Almost every deep result about axioms — consistency, independence, categoricity — is ultimately a statement about which models do or do not exist.