INFINITE SETS AND NUMBERS by Robert Bunn B.A., University of North Dakota, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Philosophy We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1974 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Philosophy The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date April 24, 1975 ABSTRACT This dissertation is a conceptual history of transfinite set theory from the earliest results until the formulation of an axio• matic set theory of Cantorian extent which avoids the paradoxes; it also contains some information on pre-Cantorian views concerning con• cepts 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 objec• tions 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 defini• tion 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 "para• doxes 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. - ii - 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 exis• tence 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 ele• ments 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 reason• ing 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 increas• ing 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 cardin• al number. CONTENTS GENERAL INTRODUCTION 1 CHAPTER I. Concepts of Finite and Infinite 8 1. Introduction 8 2. The Mathematical Concept of Infinity 13 3. Dedekind's Definition 29 4. Potential Infinity 34 5. Cantor on the Actual and Potential Infinite 37 6. Absolute Infinity 41 CHAPTER II. Infinite Numbers 44 7. Traditional Objections to Infinite Numbers 44 8. Abstract Number and Numerical Equality 50 9. The Frege-Russell Definition of Numbers 54 10. Proofs that there are Infinite Sets and Numbers 61 CHAPTER III. Quantitative Relations Between Infinite Sets. 70 11. Traditional Definitions of Greater and Less 70 12. Leibniz and the Problem of the Infinite 73 13. Bolzano on Quantitative Relations... 79 CHAPTER IV. Cantor's Theory of the Transfinite 82 14. Cantor's Definitions 82 15. Cantor's Theorem -. 83 16. Relations in Extension 87 17. The Numbers of Transfinite Weil-Ordered Sets 93 18. Cardinal and Ordinal Numbers 106 19. Proof and Definition by Transfinite Induction 118 20. Number Classes and Alephs 127 CHAPTER V. The Theory of Limitation of Size 135 21. Introduction 135 22. The Cantor-von Neumann Analysis of the Paradoxes 141 23. Russell on the Theory of Limitation of Size 152 24. Paradoxes of the Ultrafinite: Hessenberg and Zermelo 160 25. Mirimanoff's Solution of "The Fundamental Problem of Set Theory." 164 26. Von Neumann's Axiom of Limitation of Size 172 - vi - vii CHAPTER VI. The Axioms of Set Theory 176 27. The Extensional Concept of Set 176 28. Axioms Implicit in Dedekind and Cantor 184 29. The Axioms of Separation and Replacement 189 30. Zermelo's Strong Axiom of Infinity 197 31. The Axiom of Choice' 200 32. The Axiom of Foundation 206 BIBLIOGRAPHY 219 1 General Introduction This dissertation covers the Cantorian theory of the infinite, the associated paradoxes, and the way of avoiding these paradoxes— by means of a system of axioms based on an analysis of the paradoxes— originating with Cantor himself. This way of avoiding the paradoxes Is distinguished from most others, and in particular from Russell's, in being a set theory of Cantorian extent. Thus, I provide a history of the main points in the development of the theory of infinite sets and numbers beginning with the great works of Cantor and Dedekind and continuing through the formulation of a set theory of Cantorian extent which avoids the known paradoxes. This is a period of about fifty years—from about 1880 to 1930; it ends with the last works of Zermelo and von Neumann on the axiomatic theory of sets. In my exposition of the basic concepts of Cantorian set theory, I emphasize the efforts to make these more precise. In general my concern is with the elements and the foundations of the theory, rather than with the rich development of the theory which also occurred by 1930. For example, Hausdorff's theory of ordered sets appeared in the same year (1908) in which the systems of Russell and Zermelo were pub• lished. Major works by Mahlo, Sierpinski, Tarski, Ulam and many others extending considerably the theory of the transfinite also appeared by 1930, but these are beyond the scope of this work. While it is not my intention to present a general history of thought about the infinite, I do discuss earlier ideas on certain topics 2 which are important in Cantor's theory.