Brief notes from Friday 2/8/08 lecture

We began by writing out carefully the proof of proposition 2.1

The discussion then moved to interpretations and models

Definition:  An interpretation is an assignment of meaning to the undefined terms of an axiomatic system.

For example, consider the following interpretation of the incidence axioms:

• Points:  The letters A, B, and C.  Note that we are not using these to label points, but rather these are the points themselves.
• Lines: The sets {A, B}, {A, C}, and {B, C}
• The relation "lies on":  Point P lies on line L if P is an element of L  For example, point B lies on lines {A,B} and {B, C}

We saw this interpretation in Thursday's lecture, and also noticed that, with this interpretation, all three incidence axioms are satisfied.  When this happens,

Definition:  A model of an axiomatic system is an interpretation for which all of the axioms of the system are true.  The interpretation given above is a model of the incidence axioms.

One of the important consequences of a model is that whatever we can prove in the axiomatic system must be true of any model of the axiomatic system.  For example, notice that the model given above contains no parallel lines (we say that this model has the elliptic property).  This tells us that there can not be a proof of Euclid Proposition 5 from the incidence axiomx.

Another interpretation which is not a model is the following:

• Points are the usual points on the surface of the sphere
• Lines are great circles on the surface of the sphere
• "lies on" has the usual meaning.

Incidence axioms 2 and 3 are clearly true of this model.  Incidence axiom 1, however, is not (an infinity of great circles pass through antipodal points). 

More on interpretations and models Monday