Set Theory and Logic

Robert R. Stoll

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486139647
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Set Theory and Logic is the result of a course of lectures for advanced undergraduates, developed at Oberlin College for the purpose of introducing students to the conceptual foundations of mathematics. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. One of the most complex and essential of modern mathematical innovations, the theory of sets (crucial to quantum mechanics and other sciences), is introduced in a most careful concept manner, aiming for the maximum in clarity and stimulation for further study in set logic.
Contents include: Sets and Relations — Cantor's concept of a set, etc.
Natural Number Sequence — Zorn's Lemma, etc.
Extension of Natural Numbers to Real Numbers
Logic — the Statement and Predicate Calculus, etc.
Informal Axiomatic Mathematics
Boolean Algebra Informal Axiomatic Set Theory Several Algebraic Theories — Rings, Integral Domains, Fields, etc.
First-Order Theories — Metamathematics, etc.
Symbolic logic does not figure significantly until the final chapter. The main theme of the book is mathematics as a system seen through the elaboration of real numbers; set theory and logic are seen s efficient tools in constructing axioms necessary to the system.
Mathematics students at the undergraduate level, and those who seek a rigorous but not unnecessarily technical introduction to mathematical concepts, will welcome the return to print of this most lucid work.
"Professor Stoll... has given us one of the best introductory texts we have seen." — Cosmos.
"In the reviewer's opinion, this is an excellent book, and in addition to its use as a textbook (it contains a wealth of exercises and examples) can be recommended to all who wish an introduction to mathematical logic less technical than standard treatises (to which it can also serve as preliminary reading)." — Mathematical Reviews.

CHAPTER 1
Sets and Relations

THE THEORY OF SETS as a mathematical discipline originated with the German mathematician G. Cantor (1845-1918). A complete account of its birth and childhood is out of the question here, since a considerable knowledge of mathematics is a prerequisite for its comprehension. Instead, we adopt the uneasy compromise of a brief sketch of these matters. If this proves too difficult for the reader, nothing is lost; on the other hand, if it is at least partially understood, something may be gained.

Cantor’s investigation of questions pertaining to trigonometric series and series of real numbers led him to recognize the need for a means of comparing the magnitude of infinite sets of numbers. To cope with this problem, he introduced the notion of the power (or size) of a set by defining two sets as having the same power if the members of one can be paired with those of the other. Since two finite sets can be paired if and only if they have the same number of members, the power of a finite set may be identified with a counting number. Thus the notion of power for infinite sets provides a generalization of everyday counting numbers. Cantor developed the theory, including an arithmetic, of these generalized (or transfinite) numbers and in so doing created a theory of sets. His accomplishments in this area are regarded as an outstanding example of mathematical creativity.

Cantor’s insistence on dealing with the infinite as an actuality—he regarded infinite sets and transfinite numbers as being on a par with finite sets and counting numbers—was an innovation at that time. Prejudices against this viewpoint were responsible for the rejection of his work by some mathematicians, but others reacted favorably because the theory provided a proof of the existence of transcendental numbers. Other applications in analysis and geometry were found, and Cantor’s theory of sets won acceptance to the extent that by 1890 it was recognized as an autonomous branch of mathematics. About the turn of the century there was some change in attitude with the discovery that contradictions could be derived within the theory. That these were not regarded as serious defects is suggested by their being called paradoxes— defects which could be resolved, once full understanding was acquired. The problems posed by Cantor’s theory, together with its usefulness, gradually created independent inter...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Preface
• Contents
• Chapter 1 Sets and Relations
• Chapter 2 The Natural Number Sequence and its Generalizations
• Chapter 3 The Extension of the Natural Numbers to the Real Numbers
• Chapter 4 Logic
• Chapter 5 Informal Axiomatic Mathematics
• Chapter 6 Boolean Algebras
• Chapter 7 Informal Axiomatic Set Theory
• Chapter 8 Several Algebraic Theories
• Chapter 9 First-Order Theories
• References
• Symbol and Notation
• Author Index
• Subject Index