categoricity

Some axiom systems are loose, describing many genuinely different worlds; others are tight, describing only one. An axiom system is categorical when all of its models are essentially the same — isomorphic, meaning you can match up their points so perfectly that every axiom-relation is preserved both ways. A categorical system does not merely constrain its subject; it pins it down completely. There is, up to relabelling, exactly one structure satisfying it, and so the axioms can rightly be said to describe that one thing.

Euclidean geometry, with Hilbert's full set of axioms including both continuity axioms, is categorical: every model of it is isomorphic to the ordinary coordinate plane of real-number pairs. The two continuity axioms are doing the decisive work here. Without them, the axioms have many non-isomorphic models — for instance a 'rational plane' using only rational coordinates, which satisfies incidence, order, and congruence but is full of gaps and is genuinely a different structure. It is completeness that forces the line to be the full real number line and thereby collapses all the variety down to a single shape. This is what licenses the phrase 'the Euclidean plane' with a definite article: the axioms determine it uniquely.

Categoricity is the strongest kind of success an axiom system can have, and it should be held apart from consistency and independence, which are about whether models exist and whether axioms are redundant. There is a subtle technical point worth flagging honestly: full categoricity here relies on the completeness axiom being a 'second-order' statement, quantifying over all sets of points. If one restricts to first-order logic, no theory can be categorical across all infinite sizes (a consequence of the Lowenheim-Skolem theorems), so the cleanest version of 'the real plane is the unique model' lives in the second-order setting. Within that setting, though, the result is exact and beautiful: Hilbert's axioms describe one world and no other.