Basic Set Theory

by Azriel Levy

Overview

Although this book deals with basic set theory (in general, it stops short of areas where model-theoretic methods are used) on a rather advanced level, it does it at an unhurried pace. This enables the author to pay close attention to interesting and important aspects of the topic that might otherwise be skipped over. Written for upper-level undergraduate and graduate students, the book is divided into two parts. The first covers pure set theory, including the basic notions, order and well-foundedness, cardinal numbers, the ordinals, and the axiom of choice and some of its consequences. The second part deals with applications and advanced topics, among them a review of point set topology, the real spaces, Boolean algebras, and infinite combinatorics and large cardinals. A helpful appendix deals with eliminability and conservation theorems, while numerous exercises supply additional information on the subject matter and help students test their grasp of the material. 1979 edition. 20 figures.

Product Details

ISBN-13:	9780486420790
Publisher:	Dover Publications
Publication date:	08/13/2002
Series:	Dover Books on Mathematics Series
Edition description:	Revised
Pages:	416
Product dimensions:	6.10(w) x 9.10(h) x 0.90(d)

Table of Contents

Part A. Pure Set Theory
  Chapter I. The Basic Notions
    1. The Basic Language of Set Theory
    2. The Axioms of Extensionality and Comprehension
    3. Classes, Why and How
    4. Classes, the formal Introduction
    5. The Axioms of Set Theory
    6. Relations and functions
  Chapter II. Order and Well-Foundedness
    1. Order
    2. Well-Order
    3. Ordinals
    4. Natural Numbers and finite Sequences
    5. Well-Founded Relations
    6. Well-Founded Sets
    7. The Axiom of Foundation
  Chapter III. Cardinal Numbers
    1. Finite Sets
    2. The Partial Order of the Cardinals
    3. The Finite Arithmetic of the Cardinals
    4. The Infinite Arithmetic of the Well Orderd Cardinals
  Chapter IV. The Ordinals
    1. Ordinal Addition and Multiplication
    2. Ordinal Exponentiation
    3. Cofinality and Regular Ordinals
    4. Closed Unbounded Classes and Stationery Classes
  Chapter V. The Axiom of Choice and Some of Its Consequences
    1. The Axiom of Choice and Equivalent Statements
    2. Some Weaker Versions of the Axiom of Choice
    3. Definable Sets
    4. Set Theory with Global Choice
    5. Cardinal Exponentiation
Part B. Applications and Advanced Topics
  Chapter VI. A Review of Point Set Topology
    1. Basic concepts
    2. Useful Properties and Operations
    3. Category, Baire and Borel Sets
  Chapter VII. The Real Spaces
    1. The Real Numbers
    2. The Separable Complete Metric Spaces
    3. The Close Relationship Between the Real Numbers, the Cantor Space and the Baire Space
  Chapter VIII. Boolean Algebras
    1. The Basic Theory
    2. Prime Ideals and Representation
    3. Complete Boolean Algebras
    4. Martin's Axiom
  Chapter IX. Infinite Combinatorics and Large Cardinals
    1. The Axiom of Constructibility
    2. Trees
    3. Partition Properties
    4. Measurable Cardinals
Appendix X. The Eliminability and Conservation Theorems
  Bibliography; Additional Bibliography; Index of Notation; Index
Appendix Corrections and Additions

Editorial Reviews

This advanced mathematics textbook introduces the theorems and proofs of cardinal numbers, the ordinals, the axiom of choice, the space of real numbers, Boolean algebras, trees, and partitions. Originally published by Springer Verlag in 1979, this reprint adds an appendix of corrections. Annotation c. Book News, Inc., Portland, OR