THE FOUNDATIONS OF GEOMETRY 



27 



6. There is at least one line. 



We are now in a position to develop considerably more theory. By 

 6 and 4 and 1 there must be at least two lines which by 3 meet in a 

 point A. Hence there must be four points at least, (B, G, D,E) which 

 do not lie in the same line. For if D were in the line BG, by 2, the 

 lines AB and AD would be the same, which is contrary to hypothesis. 

 A set of four points, such as A, B, G, D, of which no three are col- 

 linear, when taken together with the lines (called the sides) joining 

 the six pairs of points, AB, BG, CD, DA, AG, BD, is called a complete 

 quadrangle. In the diagram below, the vertices of a complete quad- 

 rangle are 0, 1, 4, 6. The three additional points 2, 3, 5, in which the 

 sides of the quadrangle intersect, are called the diagonal points. 



We have shown our axioms sufficient to establish the existence of a 

 complete quadrangle; are they sufficient to prove the ordinary prop- 

 erties of such a figure? They are not. Axioms 1-6 do not decide 

 whether the three diagonal points, 2, 3, 5, are or are not collinear. In 

 the ordinary geometry, those points are non-collinear and form what 

 is called the diagonal triangle. If, however, we suppose that they 

 are collinear (one may assist one's imagination by means of the dotted 

 line) then on rereading our six postulates they will all be found verified. 

 In order to obtain the usual geometry it is necessary to assume as an 



axiom that the diagonal points of a 

 complete quadrangle are non-collinear. 

 What we have just done is a simple 

 case of an ' independence proof.' We 

 have proved that the proposition that 

 the diagonal points of a complete 

 quadrangle are not collinear, is inde- 

 pendent of propositions 1, • • • 6, that 

 is, it is not a logical consequence of 

 them. Similarly, the non-Euclidean 

 geometry is an independence proof 

 for Euclid's axiom 12. The ideal of students of foundations of geom- 

 etry is a system of axioms every one of which is independent of all the 

 rest. To attain this ideal it is necessary to construct for each axiom an 

 example in which it is untrue while all the rest are verified. 



After seeing the bizarre construction that this process gives rise 

 to, one is tempted to raise the question, how can we be sure that the 

 complete system which we use applies uniquely to the space of our 

 intuition or experience and not also to one of these mathematical 

 dreams? In answering this question we define what is meant by a 

 categorical system of axioms. 



Eeturning to our complete quadrangle with collinear diagonal points 



Fig. 5. 



