Quantifier-Free Axioms For Constructive Affine Plane Geometry

The purpose of this paper is to state a set of axioms for plane geometry which do not use any quantifiers, but only constructive operations. The relevant definitions and general theorems are stated; for reasons of space the proofs are only sketched. Quantifier-free arithmetic and quantifier-free algebra have been the subjects of several investigations, beginning at least with the early important work of Herbrand. Quantifier-free axioms for plane geometry have received less attention. In a way, this is surprising, for an emphasis on geometric constructions is a significant aspect of ancient Greek geometry. The spirit of this article is that of Moler and Suppes (1968), who give quantifier-free axioms for the two constructions of finding the intersection of two lines and of laying off one segment on another. The representation theorem for models of their constructive theory is in terms of vector spaces over Pythagorean fields, a geometry discussed by Hilbert in his Foundations of Geometry but not axiomatized. The present work has two important differences. First, the emphasis is on finite configurations, rather than on closure of the constructions to yield a representation theorem in terms of a vector space over the ordered field of rationals. Second, the affine axioms, though numerous, are individually much simpler and avoid some troublesome problems corresponding to division by zero. The axioms are summarized without comment in the appendix. The axioms are clearly not independent; for example, those on linearity can be derived from the later order axioms.