The axiomatics of Euclidean geometry.

Abstract

The main object of this thesis is to provide axiomatizations for Euclidean geometry, that are, in some precisely defined sense, simpler than any other imaginable axiomatization thereof. After stating some simplicity criteria, we propose axiom systems for several fragments of Euclidean geometry that are most simple according to some of these criteria. Some of the axiomatizations are carried out in first-order logic, others in first-order logic with all axioms universal, and others in algorithmic logic. It is shown, for example, that both 2-dimensional and 3-dimensional Euclidean geometry over Euclidean ordered fields $({\cal E}\sbsp{2}{\prime}$ and ${\cal E}\sbsp{3}{\prime})$ can be axiomatized in a language with two predicates, standing for 'congruence' and 'betweenness', by an axiom system all of whose axioms contain, when written in prenex form, at most 5 variables. In this particular language, this axiomatization is best possible, in the sense that it uses the smallest number of variables in its axioms. For three dimensional Euclidean geometry it is best possible, regardless of language. This result is improved to show that, for theories that are definitionally equivalent to a wide variety of fragments of ${\cal E}\sbsp{2}{\prime}$, we have axiom systems that are expressed in languages without relation symbols (i.e., they contain only operation symbols), by universal axioms, each axiom containing at most 4 variables. This is best possible, regardless of language. It is also shown that the languages in which these theories are expressed are as simple as possible, in the sense that all operations are at most ternary. It is known that one cannot axiomatize geometry by using only binary geometric operations. Other results include the splitting of Euclid's parallel postulate into two strictly weaker geometric statements, and an infinitary axiomatization of geometry with a unit distance in a language with only one binary relation symbol.