ABSTRACT
This chapter starts a formal development of the foundations of analysis, beginning with an axiomatic treatment of the real numbers. Logic provides our model for such a development. The essential building blocks of the subject, coming from intuition and vast experience, are presented as axioms. Except for the foundational axioms, results are incorporated into the mathematical edifice only when they are proven. The axioms describing the properties of the real numbers fall into
three categories: field axioms, order axioms, and completeness axioms. The more elementary field and order axioms will be presented first, along with some of their immediate consequences. The subsequent addition of the completeness property marks something of a shift in the character of the subject. It is here that the infinite processes, viewed with suspicion by the ancient Greeks, come into play. Three versions of the completeness property of the real numbers will be considered. The completeness property is an invaluable tool for working with
the infinite sequences that arise so commonly in analysis. Following the initial study of completeness, the compactness property for closed bounded intervals [a, b] is introduced. Completeness also plays a role in the concluding topics of the chapter, infinite products and continued fractions. The treatment of proofs in propositional logic will serve as a guide
as we move beyond the axioms. Conjectures about ideas and results that might be true can be inspired by examples, intuition, or brilliant guesswork. However, acceptance of such results will only come from rigorous proofs. Proofs should consist of a careful and complete sequence of arguments; each step should either be previously established, or be a logical consequence of previously established results. To avoid getting hopelessly bogged down in technical details, a con-
siderable amount of foundational material is assumed to be known in advance. Some of this material includes properties of sets, functions, the equality predicate =, and the elementary properties of the integers
will also be exploited. The formal treatment of mathematics, with its emphasis on careful
proofs, is a very cautious and sometimes difficult approach, but the resulting structure has a durability and reliability rarely matched in other subjects. The choice of the real numbers as the focus for axiomatic characterization is efficient, but it should be noted that there are alternative treatments which place the emphasis on the integers and rational numbers. Such alternatives, which define real numbers in terms of rational numbers, are attractive because they minimize the assumptions at the foundations of mathematics. Such an alternative development may be found in [16, pp. 1-13] and in [18, pp. 35-45].
