Dimension in Elementary Euclidean Geometry

Abstract

The convention adopted in this chapter is that usual geometrical notions do not involve infinite sets or infinite sequences of points. In specific, a standard Euclidean space can be a vector space over the field of real numbers having any finite or infinite linear dimension. Geometrical properties of spaces are formulated in terms of geometrically meaningful notions. The elementary geometrical properties are identified with the properties of a space expressible in sentences of the first-order predicate logic in terms of the geometrical relations over the space. The chapter presents specific geometric results and a general theorem in the theory of models of the first-order logic. This general result is then applied to geometry and it is shown that there are no elementary geometrical properties distinguishing any two infinite dimensional Euclidean spaces.