Categorical Axiomatic System

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.

One way to see that this is a non-categorical axiomatic system is to note that one can form a compatible system from two fundamentally different models, e.g.,

1. Two disjoint lines each containing two points plus a separate point not on either line.

2. Two lines containing three points each which intersect in one of the points.

The presence of an intersection in one model and not the other implies that the models are fundamentally different and hence are inequivalent.

REFERENCES

Smart, J. "Finite Geometries and Axiomatic Systems." 2002. http://www.beva.org/math323/asgn5/nov5.htm.