Set Theory and the Continuum Hypothesis

Paul J. Cohen

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486134710
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. The independence of the continuum hypothesis is the focus of this study by Paul J. Cohen. It presents not only an accessible technical explanation of the author's landmark proof but also a fine introduction to mathematical logic. An emeritus professor of mathematics at Stanford University, Dr. Cohen won two of the most prestigious awards in mathematics: in 1964, he was awarded the American Mathematical Society's Bôcher Prize for analysis; and in 1966, he received the Fields Medal for Logic.
In this volume, the distinguished mathematician offers an exposition of set theory and the continuum hypothesis that employs intuitive explanations as well as detailed proofs. The self-contained treatment includes background material in logic and axiomatic set theory as well as an account of Kurt Gödel's proof of the consistency of the continuum hypothesis. An invaluable reference book for mathematicians and mathematical theorists, this text is suitable for graduate and postgraduate students and is rich with hints and ideas that will lead readers to further work in mathematical logic.

CHAPTER I

GENERAL BACKGROUND IN LOGIC

1. INTRODUCTION

It is now known that the truth or falsity of the continuum hypothesis and other related conjectures cannot be determined by set theory as we know it today. This state of affairs regarding a classical and presumably well-posed problem must certainly appear rather unsatisfactory to the average mathematician. One is tempted to look more closely and perhaps more critically at the foundations of mathematics. Although our present “Cantorian” mathematics is highly successful in its treatment of abstractions, one must not overlook the fact that from the very beginning the use of infinite processes was regarded with suspicion by many people. In the 19th century, the objections regarding the use of convergent series and real numbers were met by Cauchy, Dedekind, Cantor and others, only to be followed by more profound criticisms from later mathematicians such as Brouwer, Poincaré and Weyl. The controversy which followed resulted in the formation of various schools of thought concerning the foundations. It is safe to say that no attitude has been completely successful in answering the fundamental questions, but rather that the difficulties seem to be inherent in the very nature of mathematics. Despite the fact that the continuum hypothesis is a very dramatic example of what might be called an absolutely undecidable statement (in our present scheme of things), Gödel’s incompleteness theorem still represents the greatest obstacle to a satisfactory philosophy of mathematics. These fundamental difficulties, often dismissed by mathematicians, make the independence of the continuum hypothesis less surprising.

Gauss seems to have been the first mathematician to have expressed doubts about too free a use of infinities. In 1831, he wrote, “I protest ...against the use of an infinite magnitude as something completed, which is never permissible.” Later, Kronecker expressed views which were critical of definitions that required an infinite process to verify that an object satisfied them. Cantor’s work on set theory was the subject of much criticism to the effect that it dealt with fictions. Nevertheless, infinite sets are accepted today with few reservations. The traditional attitude accepts the construction of the real number system from the rationals as the last and final step in the long series of criticisms and re-examinations which have marked the history of mathematics. What possible objections can be raised to the construction of the real numbers? Simply this: although the reals are based on the integers, the vague notion of an arbitrary set of integers (or, equivalently, an arbitrary sequence of integers) must be introduced. Mathematicians inclined to a finitist point of view might hold that only sets which have an explicit rule to determine which integers are in the set should be admitted. For example, the school of Brouwer (Intuitionism) would only admit finite sets as legitimate objects of study, and even a single integer would not be considered defined unless a very definite rule for computing it was given. (For instance, the set consisting of 5 if Fermat’s Last Theorem is true and 7 if it is false is not well-defined, according to Brouwer.) The criticism of Weyl and Poincaré was directed against “impredicative” definitions. Although their objections were not as extreme as Brouwer’s, the acceptance of these criticisms would mean the destruction of large portions of mathematics.

Another source of objections was the paradoxes or antinomies of set theory. In Cantor’s set theory a set was thought of as being defined by a property. Cantor himself pointed out that the set of all sets leads to an absurdity. Although this type of paradox (along with those of Russell, Burali-Forti and others), seems entirely remote from ordinary mathematical reasoning, the paradoxes did point out the necessity of extreme care when attempting to describe which properties describe sets. In 1908, Zermelo presented a formal set of axioms for set theory which encompassed all the present day reasonings in mathematics and yet which is presumably free from paradoxes. This axiomatization of set theory was in keeping with the spirit of the school of Formalism, led by David Hilbert. According to the Formalist point of view, mathematics should be regarded as a purely formal game played with marks on paper, and the only requirement this game need fulfill is that it does not lead to an inconsistency. To completely describe the game required setting down the rules of mathematical logic with much greater precision than had been previously done. This was done, and the Formalists turned their attention to showing that various systems were consistent. As is well known, this hope was destroyed by Gödel’s discovery of the incompleteness theorem, which implies that the consistency of a mathematical system cannot be proved except by methods more powerful than those of the system itself.

Despite this failure, the Formalist program contributed greatly to the development of logic by establishing a systematic study of mathematical systems. In these notes, our first object will be to describe how a mathematical system can be completely reduced to a purely formal game involving the manipulation of symbols on paper. By a formal system we shall mean a finite collection of symbols and perfectly precise rules for manipulating these symbols to form certain combinations called “theorems”. Of course, these rules must be given in informal mathematical language. However, we shall demand that they be completely explicit rules requiring no infinite processes to check and that in principle they can be coded into a computing machine. In this way questions concerning infinite sets are replaced by questions concerning the combinatorial possibilities of a certain formal game. Then we will be able to say that certain statements are not decidable within given formal systems.

2. FORMAL LANGUAGES

If we examine Peano’s axioms for the integers, we find that they are not capable of being transcribed in a form acceptable to a computing machine. This is because the crucial axiom of induction speaks about “sets” of integers but the axioms do not give rules for forming sets nor other basic properties of sets. Here is an example of the difficulty in satisfying the stringent requirements for a formal system outlined above. When we do construct a formal system corresponding to Peano’s axioms we shall find that the result can not quite live up to all our expectations. This difficulty is associated with any attempt at formalization.

Two types of symbols will appear in our formal language. First, there are those symbols common to all mathematical systems. Then there are those used to denote particular concepts in special branches of mathematics, such as the symbols for addition, group multiplication, adjoint of a matrix, etc. The general symbols we shall use are the following:

The words beneath the symbols have, in principle, nothing to do with our formal language. Rather, we are t...

Table of contents

• Cover
• Dover Books on Mathematics
• Title Page
• Copyright Page
• Foreword
• Introduction to the Dover Edition
• My interaction with Kurt Gödel: The man and his work
• Preface
• Table of Contents
• Chapter I. General background in logic
• Chapter II. Zermelo-Fraenkel set theory
• Chapter III. The Consistency of the continuum hypothesis and the axiom of choice
• Chapter IV. The independence of the continuum hypothesis and the axiom of choice
• References