An axiomatic system is said to be categorical if there is only one essentially distinct representation for it. In particular, the names and types of objects within the system may vary while still being considered "the same, " e.g., geometries and their plane duals. An example of an axiomatic system which isn't categorical is a geometry described by the following four axioms (Smart): 1. There exist five points. 2. Each line is a subset of those five points. 3. There exist two lines. 4. Each line contains at least two points.

axiom | geometry | intersection | line | point