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

by Keene, G. B.

• ISBN13: 9780486462493
• ISBN10: 0486462498
• Format: Paperback
• Copyright: 2007-12-10
• Publisher: Dover Publications

Summary

This text unites the logical and philosophical aspects of set theory in a manner intelligible both to mathematicians without training in formal logic and to logicians without a mathematical background. It combines an elementary level of treatment with the highest possible degree of logical rigor and precision. 1961 edition.

Table of Contents

Preface	p. v
The Elements of Set Theory
The Basic Logical Concepts
Introduction	p. 3
The logical constants	p. 3
The use of quantifiers in proofs	p. 13
Operations on Classes
Classes and composite wholes	p. 16
Conventions for diagramming class-structure	p. 17
Elementary operations	p. 22
The Bernays Theory of Finite Classes and Finite Sets
Foreword	p. 35
Introduction
Pairs, ordered pairs and k-tuplets	p. 36
Class conditions	p. 38
The Basis of the System
Primitive constants	p. 43
Initial definitions	p. 44
The axioms of the system	p. 45
The classes admitted by AxIII	p. 46
The Development of the System: Stage I
Stage I theorems	p. 48
Standard expressions and the class theorem	p. 51
The Development of the System: Stage II
Stage II theorems	p. 64
Fundamental theorems on ordinals	p. 66
The Theory of Finite Ordinals
Finite ordinal number	p. 80
The Peano axioms as theorems of the system	p. 81
Iteration	p. 82
The existence theorem for the function [characters not reproducible]	p. 87
Addition, multiplication and exponentiation	p. 89
The Theory of Finite Classes and Finite Sets
Preliminary theorems on finite ordinals	p. 91
Theorems on finite classes and finite sets	p. 99
Conclusion	p. 101
References	p. 103
Index	p. 105
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