Mathematical Systems

A mathematical system  consists of the following:

1. a set of undefined concepts,
2. a universal set,
3. a set of relations,
4. a set of operations,
5. a set of logical axioms,
6. a set of non-logical axioms--these axioms pertain to the elements being studied, the relations, and the operations; and not to the logic being used,
7. a set of theorems,
8. a set of definitions,
9. an underlying set theory.

In plane geometry the undefined concepts were those of point and line. The universal set was the set of points in the plane. The relations were such concepts as equality, perpendicularity, and parallelism. We have mentioned the logical axioms. A non-logical axiom would be of the form:

Two different points are on exactly one line.