What is Elementary Geometry?

Abstract

This chapter describes the significance of notions and methods of modern logic and metamathematics for the study of the foundations of geometry. It focuses on the conception of elementary geometry, which can be described as the part of Euclidean geometry that can be formulated and established without the help of any set-theoretical devices. Elementary geometry is formalized within elementary logic that is essentially first-order predicate calculus. The logical constants of the theory include the sentential connectives, the quantifiers, and the two special binary predicates. As non-logical constants any predicates can be chosen denoting certain relations among points in terms of which all geometrical notions are known to be definable. In the formalization of elementary geometry, only points are treated as individuals and are represented by first-order variables. Elementary geometry has no set-theoretical basis, because of which its formalization does not provide for variables of higher orders and no symbols are available to represent or denote geometrical figures and classes of geometrical figures.