An Axiomatic Development of the Properties of Real Numbers Through the Use of Set Theory

The purpose of this paper is to present a logical development of the real number system from a few basic assumptions in a form that an undergraduate can use. The development is based on properties of sets and begins with cardinal and ordinal numbers. It then proceeds through the natural numbers, denumerable sets, rational numbers, and real numbers. The rational numbers are developed in three steps: (1) non-negative fractions, (2) non-negative rational numbers, and (3) the full set of rational numbers. The distinction between these three divisions of the rational numbers is explained fully. Chapter X concludes the study with a discussion of the “axiom of choice” and entity closely related to this study