Models

One way of checking the consistency of our system of axioms is to construct ``models'' for which all the axioms are verified. Of course, these verifications again use results from some other area of mathematics and the axioms of that would also have to be verified to be consistent and so on. This is the idea behind the impossibility of verifying consistency. Leaving philosophical studies behind let us examine ``three dimensional projective geometry over a (skew-)field''.

Exercise 4   Let K be a skew-field (i.e. K has addition, subtraction, multiplication and division but multiplication does not necessarily commute). Points, lines and planes of P³(K) are given by (left) linear subspaces of K⁴ of rank 1, 2 and 3 respectively. The incidence relations are just the inclusions of subspaces. Show that this gives a system that satisfies the above Incidence axioms and the projective axiom of parallels.

In fact, this even leads to another system which satisfies the usual axiom of parallels.

Exercise 5   Let A³(K) be the collection of all points, lines and planes in P³(K) that are not contained in a fixed plane $\alpha$ (called the plane ``at the horizon''). Show that this geometry satisfies all the axioms of incidence and the ``usual'' axiom of parallels.

The notion of between-ness can also be brought in with some more algebra.

Definition 1   A positivity on K is a subset P so that:

1. P + P $\subset$ P and P ^. P $\subset$ P.
2. P $\cup$ {0} $\cup$ (- P) = K and this is a disjoint union.

This conforms to the concept of positive numbers. Using this we can define the cone generated by a collection of vectors in K⁴ as the collection of all non-negative linear combinations of the vectors.

Exercise 6   Fix a three dimensional linear subspace V of K⁴ (in other words a plane in P³(K)) and a vector v not in V. There is a unique linear functional on K⁴ which has kernel V and takes the value 1 on v. We say a vector w is positive if f (w) lies in P. Every linear subspace in K⁴ which does not line in V is then determined by its positive half.

Exercise 7   We say that a point A of A³(K) lies between points B and C if the positive half of the linear subspace in K⁴ corresponding to A is a positive linear combination of of the positive halves of the linear subspaces corresponding to B and C respectively. Check that the axioms of order are satisfied on A³(K) with this notion of between-ness.

We have thus constructed a geometry satisfying all our axioms by making use of some algebra. Other geometries satisfying these axioms can also be constructed.

Definition 2   A collection R of points in A³(K) is said to be convex if, given A and B are points in R and C in A³(K) is between A and B, then C is also in R.

Definition 3   A convex collection R of points is said to be open if for any point A in R and B in A³(K), there is a point C lying between A and B in A³(K) so that C is also in R.

Exercise 8   Let R be an open convex collection of points in A³(K). We denote by [R] the geometry for which points are the points of R, lines and planes of [R] are the lines and planes of A³(K) which meet R. The relations of incidence and order are inherited from A³(K). Check that this geometry satisfies the axioms of incidence and order.

A very important result (a sketch of proof is outlined in the next section) is that every geometry satisfying the axioms of incidence and order is of the type [R] for an open convex set R in A³(K) for a suitable ordered field K.

Hence, and this is important to note, the fact that arithmetic/algebraic problems arise in geometry does not immediately have anything to do with measurement! In particular, the relation between distance and coordinates can be much more complicated than that which will emerge from the Pythagoras theorem.