Infinite sets and numbers

This dissertation is a conceptual history of transfinite set theory from the earliest results until the formulation of an axiomatic set theory of Cantorian extent which avoids the paradoxes; it also contains some information on pre-Cantorian views concerning concepts important in Cantor's theory. I begin with explications of the concept of infinite set due to Dedekind, Pierce, Frege, and Russell. The initial chapter features Dedekind's work, since he was the first to give rigorous demonstrations involving the concepts "finite" and "infinite." Chapter Two describes traditional views on infinite numbers, as well as on numbers in general. It emerges that the traditional objections against infinite numbers were based merely on the fact that numbers had been defined to be finite. I also discuss the Frege-Russell definition of the cardinal numbers, which was the first precise definition of numbers to accommodate infinite as well as finite numbers, and I analyze the proofs that there are infinite sets and numbers. Chapter Three deals with pre-Cantorian views on quantitative relations between infinite sets, and the associated old-fashioned "paradoxes of the infinite." I find that the errors of pre-Cantorian authors regarding quantitative relations between infinite sets were due to the mistaken belief that the relations of greater and less, when defined in the traditional way, are incompatible with numerical equality defined as one-one correspondence. The chapter on the Cantorian theory of the transfinite discusses the basic concepts and the main theorems which either were proved by Cantor or are generalizations of theorems proved by Cantor. Cantor himself did not present the concepts of the theory of the transfinite in the most precise way, and often formulated definitions (and theorems) for the ordinals of the first two number classes which can be extended to ordinals in general. I describe improvements and generalizations introduced by Russell, Hausdorff, von Neumann, and others. While developing the theory of the transfinite, Cantor came upon several paradoxes, and other mathematicians and logicians discovered more paradoxes later. My last two chapters deal with the analysis of these paradoxes which originated with Cantor, and with the corresponding way of avoiding them. According to Cantor's analysis of the paradoxes, the properties which do not determine classes are exactly those which belong to as many things as there are in some 'absolute totality' such as the totalities of all ordinals, all sets, or all entities. There are, for example, at least as many classes which do not belong to themselves as there are ordinals, as was shown by Russell. The way of avoiding the paradoxes corresponding to Cantor's analysis is a system of axioms which implies the theorems of the theory of the transfinite, but not the existence of 'absolute totalities' or totalities of equal power. Cantor himself formulated several important axioms in correspondence with Dedekind. Later, Zermelo published a system of axioms in accordance with the idea that the 'paradoxical classes' are those which are 'too big,' and he showed that some of the main theorems of Cantor's theory of transfinite cardinals can be derived from these axioms. Von Neumann formulated a system of axioms based on the idea that in the case of properties (e.g. the property of being a class) which do not determine classes, some of the classes having such properties are not elements of other classes; therefore there can only be a class of all elements having such a property (e.g. a class of all classes which are elements), and such classes are not elements. Von Neumann's system included an axiomatic criterion for being a class which is an element, i.e. a set: A class is a set if and only if it is not of at least the power of the class of all elements. The axioms formulated by Cantor, as well as the axiom of choice, are theorems of the system containing this axiom. Later, von Neumann showed that his axiom of limitation of size follows from a system containing the axioms of choice, replacement, and foundation. It follows from the axiom of foundation and von Neumann's theory of limitation of size that the universe of elements decomposes into a sequence of (disjoint) strata containing sets of ever greater complexity, which sequence is similar to the sequence of ordinals. The classes that are not elements are the classes containing sets from these strata but not themselves belonging to any of the strata, since they contain elements from "too many" of the strata in the sense that, for any stratum, they contain elements of higher strata. In general, my investigations show the subject of quantitative relations to be an important and pervasive factor in the history of the theory of the transfinite. A great deal of widespread erroneous reasoning about the infinite concerned quantitative relations. Some of the principle mathematical problems of Cantor's theory concerned these relations: Are there unequal transfinite powers? Is there an increasing sequence of transfinite powers? Are any two transfinite powers comparable? The new difficulties in the theory of the transfinite were discovered by attempting to solve these problems and by reflection on the solutions. Analysis of the paradoxes seems to show that the classes involved in the paradoxes are those which would have the greatest cardinal number.Arts, Faculty ofPhilosophy, Department ofGraduat