Axiomatic Systems for Geometry∗

An axiomatic system contains a set of primitives and axioms. The primitives are Adaptation to the current course is in the margins. object names, but the objects they name are left undefined. This is why the primitives are also called undefined terms. The axioms are sentences that make assertions about the primitives. Such assertions are not provided with any justification, they are neither true nor false. However, every subsequent assertion about the primitives, called a theorem, must be a rigorously logical consequence of the axioms and previously proved theorems. There are also formal definitions in an axiomatic system, but these serve only to simplify things. They establish new object names for complex combinations of primitives and previously defined terms. This kind of definition does not impart any ‘meaning’, not yet, anyway. See Lesson A6 for an elaboration. If, however, a definite meaning is assigned to the primitives of the axiomatic system, called an interpretation, then the theorems become meaningful assertions which might be true or false. If for a given interpretation all the axioms are true, then everything asserted by the theorems also becomes true. Such an interpretation is called a model for the axiomatic system.