On a set of postulates which suffice to define a number-plane

Abstract

In this paper I propose to show that any plane satisfying Veblen's Axioms I-VIII, XI of his System of axioms for geometry^ is a number-plane, or in other words that any plane V satisfying these axioms contains a system of continuous curves such that, with reference to these curves regarded as straight lines, the plane V is an ordinary euclidean plane. In consequence, any discussion of analysis situs based on these axioms (as, for example, Veblen's proof I of the theorem that a Jordan curve divides its plane into just two parts) is no more general than one based on analytic hypotheses. This does not contradict the fact that the geometry based on axioms I-VIII, XI is much more general than euclidean geometry in the sense that the curves with respect to which the plane V is euclidean are not necessarily the straight lines referred to in these axioms. My argument also furnishes the answer to a problem proposed by Veblen in another paper. He states this problem as follows.§ An interesting situation is obtained by introducing a postulate of uniformity among the hypotheses of plane analysis situs (cf. p. 84 of this volume). If the postulate is applied to the straight line, the line is necessarily a continuum but it is not obvious that other curves are. If it is applied to the plane, the segments <ryP in this case being triangular regions, all continuous curves are continua, but it is not obvious that there is a one-to-one reciprocal correspondence between the plane and a set of number-pairs. Without adding any postulate of uniformity to Veblen's Axioms I-VIII,