Some Comments on the Role of the Axiom of Choice in the Development of Abstract Set Theory

Abstract

The axiom of choice asserts that if 41 is a set of non-null, disjoint sets mi, then there exists a set S, called a set of choice, which contains one and only one element from each of the mni. The existence of such a collection as S seems assured on the basis of intuitive reasoning, However, it must be recalled that the word set is undefined, and only attains meaning through its occurrences in the axioms of set theory. An equivalent assertion is that of the well-ordering theorem which states that every class can be so ordered that each of its non-empty subclasses has a first element. This article gives a. brief description of the significance of the axiom of choice and the well-orderingtheorem in the development of abstract set theory. The birth of Cantor's theory of sets. The use of the procedure of cuts by Dedekind [11i to define irrational numbers made it clear that the real numbers could only be attained (conceptually) by conceiving of an actual infinity of numbers previously introduced (rationals). Thus, it became necessary to admit reasoning which dared to use an actual infinity of propositions as premises, each corresponding in a one-to-one fashion to the rationals 1221. Manyof thepredecessors of Cantor had tried to reason on the infinite, but so many contradictions were met that the actual infinite was relegated to a place outside of mathematics. Cantor taught mathematicians to consider certain distinctions such as that between cardinal and ordinal, which only the grammarians had previously observed. Cantor also pointed out that, although there is no real distinction in the case of finite collections, it becomes necessary to distinguish between these conceptions in the case of infinite aggregates [41]. It can be shown that it is by means of the wellordering theorem (or, equivalently, the axiom of choice) that the concepts of cardinal and ordinal are united for the infinite case too. Many difficulties arose when analysis was made of objects whose