Abstract Sets and Finite Ordinals: An Introduction to the Study of Set Theory

PREFACE

IN this book an attempt is made to present, in a reasonably simple way, the outlines of a relatively complex subject. Although there are now many textbooks devoted to the relationship between mathematics and the logic of classes, very few of them are of much help to the newcomer. They usually presuppose either some knowledge of mathematical logic, despite the fact that few mathematics students are trained in logic, or else considerable competence in mathematics, thus neglecting the needs of many philosophy students. The present text aims to meet the needs of students both of mathematics and of philosophy, as well as those of the general reader. For this reason a preliminary survey is made, in Part I, of the few symbols of logic used in the second half of the text. It can safely be taken as read by anyone who has studied elementary modern logic. At the same time, no knowledge of mathematics is presupposed.

The basis of Part II is the System of Axiomatic Set Theory of Paul Bernays, originally published as a series of articles in the Journal of Symbolic Logic¹. In these articles Bernays makes as little explicit use as possible of the formalism of logic. But in doing so he makes implicit use of the professional reader’s background knowledge. Consequently much of the intricate reasoning which lies just below the surface, so to speak, is beyond the grasp of readers new to the subject. The purpose of this book is to present a fragment of the Bernays Theory in a version which, while it makes explicit use of a certain amount of formalism, calls for no previous acquaintance with the subject.

The result, it is hoped, will serve as an introduction both to the complete Bernays theory and to the techniques of set theory in general. It should also provide some point of application for elementary logic, less trivial than the usual testing-for-validity of artificially devised arguments.

I am indebted to Professor Bernays for his criticisms of the first draft of the book. His suggestions as well as his kindly interest in it considerably lightened the burden of final preparation and correction of the typescript. I am also indebted to Mr. P. Geach of Birmingham, who read the typescript in an early form, to Professor D. J. O’Connor of Exeter University and Professor I. N. Sneddon of Glasgow University for their help and encouragement, and to the Editor of the Journal of Symbolic Logic, Professor S. C. Kleene, for his approval of my adaptation of material published in that Journal.

Exeter

G. B. K.

PART I

THE ELEMENTS OF SET THEORY

SECTION 1

THE BASIC LOGICAL CONCEPTS

1.1. Introduction

THE symbolism of elementary logic is both simple and efficient. By its means elegant and powerful logical calculi can be set up for the purpose of formalizing a scientific theory. But familiarity with the techniques involved in setting up these logical calculi is not essential to an understanding of the present text. All that is required is an understanding of the meaning and interrelation of the following symbols:

∼, e9780486155005_img_8729.gif , v, ⊃, ≡, (x)φ, (∃x)φ, ε

All of them translate straightforwardly into a familiar word or phrase of ordinary speech. But they are not mere shorthand symbols. For each differs from its counterpart in ordinary speech in virtue of the fact that it is precisely defined; whereas the words of ordinary speech are not. We shall formulate, explain and illustrate the definition of each in turn. In doing so we shall make use of the two groups of letters: x,y,z,w, and: P,Q,R,S, which we call variables. They are variables in the sense that they keep a place open in a formula, just as the x in x² keeps a place open for any number we may wish to put there. The type of entity that may be put in place of our variables will be called the range of values of those variables. In fact the range of values of the variables x,y,z,w, is any individual object of any kind, and the range of values of the variables P,Q,R,S, is any proposition. Finally we shall use the notation T, to mean any true proposition, and F to mean any false proposition, and we shall refer to the symbols listed above as logical constants.

1.2. The Logical Constants

1.21. Member of

When the Greek letter ε occurs between two variables as in:

x ε y

the range of values of the variable y is, thereby, confined to classes. Thus if the value John Doe is given to the variable x and the value mortals to the variable y, the result is the formula:

John Doe ε mortals

which may be read:

John Doe is a member of the class Mortals

Again if we write Nn as the name of the class Natural Numbers, then the formula:

(I)

x ε Nn

gives rise to the following formulae, each of which represents a true statement:

(II)

0 ε Nn 1 ε Nn

2 ε Nn

… etc.

Any formula of the type (I) above will be called a propositional function, where this term is intended to mean: an expression which becomes a proposition (true or false) when values are