Axiomatic set theory

Table of Contents:

1. Introduction

1.1 Set Theory and the Foundations of Mathematics
1.2 Logic and Notation
1.3 Axiom Schema of Abstraction and Russell's Paradox
1.4 More Paradoxes
1.5 Preview of Axioms

2. General Developments

2.1 Preliminaries: Formulas and Definitions
2.2 Axioms of Extensionality and Separation
2.3 "Intersection, Union, and Difference of Sets "
2.4 Pairing Axiom and Ordered Pairs
2.5 Definition by Abstraction
2.6 Sum Axiom and Families of Sets
2.7 Power Set Axiom
2.8 Cartesian Product of Sets
2.9 Axiom of Regularity
2.10 Summary of Axioms

3. Relations and Functions

3.1 Operations on Binary Relations
3.2 Ordering Relations
3.3 Equivalence Relations and Partitions
3.4 Functions

4. "Equipollence, Finite Sets, And Cardinal Numbers"

4.1 Equipollence
4.2 Finite Sets
4.3 Cardinal Numbers
4.4 Finite Cardinals

5. Finite Ordinals and Denumerable Sets

5.1 Definition and General Properties of Ordinals
5.2 Finite Ordinals and Recursive Definitions
5.3 Denumerable Sets

6. Rational Numbers and Real Numbers

6.1 Introduction
6.2 Fractions
6.3 Non-negative Rational Numbers
6.4 Rational Numbers
6.5 Cauchy Sequences of Rational Numbers
6.6 Real Numbers
6.7 Sets of the Power of the Continuum

7. Transfinite Induction and Ordinal Arithmetic

7.1 Transfinite Induction and Definition by Transfinite Recursion
7.2 Elements of Ordinal Arithmetic
7.3 Cardinal Numbers Again and Alephs
7.4 Well-Ordered Sets
7.5 Revised Summary of Axioms

8. The Axiom of Choice

8.1 Some Applications of the Axiom of Choice
8.2 Equivalents of the Axiom of Choice
8.3 Axioms Which Imply the Axiom of Choice
8.4 Independence of the Axiom of Choice and the Generalized Continuum Hypothesis

One of the most pressing problems of mathematics over the last hundred years has been the question: What is a number? One of the most impressive answers has been the axiomatic development of set theory. The question raised is: "Exactly what assumptions, beyond those of elementary logic, are required as a basis for modern mathematics?" Answering this question by means of the Zermelo-Fraenkel system, Professor Suppes' coverage is the best treatment of axiomatic set theory for the mathematics student on the upper undergraduate or graduate level.
The opening chapter covers the basic paradoxes and the history of set theory and provides a motivation for the study. The second and third chapters cover the basic definitions and axioms and the theory of relations and functions. Beginning with the fourth chapter, equipollence, finite sets and cardinal numbers are dealt with. Chapter five continues the development with finite ordinals and denumerable sets. Chapter six, on rational numbers and real numbers, has been arranged so that it can be omitted without loss of continuity. In chapter seven, transfinite induction and ordinal arithmetic are introduced and the system of axioms is revised. The final chapter deals with the axiom of choice. Throughout, emphasis is on axioms and theorems; proofs are informal. Exercises supplement the text. Much coverage is given to intuitive ideas as well as to comparative development of other systems of set theory. Although a degree of mathematical sophistication is necessary, especially for the final two chapters, no previous work in mathematical logic or set theory is required.
For the student of mathematics, set theory is necessary for the proper understanding of the foundations of mathematics. Professor Suppes in Axiomatic Set Theory provides a very clear and well-developed approach. For those with more than a classroom interest in set theory, the historical references and the coverage of the rationale behind the axioms will provide a strong background to the major developments in the field. 1960 edition.


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