An axiomatic basis for plane geometry

Abstract

1. The axioms. The fourth appendix of Hilbert's Grundlagen der Geometriet is devoted to the foundation of plane geometry on three axioms pertaining to transformations of the plane into itself. The object of the present paper is to attain the same end by quicker and simpler means. The simplifications are made possible by using orientation-reversing transformationst and changing Hilbert's second and third axioms. The (x, y)-plane will mean the set of all distinct ordered pairs of real numbers. The terms of analytic geometry, to which no geometric content need be given, will be used, modified by the prefix (x, y) where ambiguity might arise. Thus we shall refer to (x, y)-distance, (x, y)-lines, and so on. The general plane, p, will be any set of objects, called points, which can be put in one-to-one correspondence with the points of the (x, y)-plane. For convenience, we shall speak of the points of p as if they were identical with their images under such a correspondence. The following axioms pertain to a set, T, of continuous? one-to-one transformations of p into itself. A transformation of the set which leaves two distinct points fixed and reverses orientation will be called a reflection.