Synopses & Reviews
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.
Synopsis
The first part of this advanced-level text covers pure set theory, and the second deals with applications and advanced topics (point set topology, real spaces, Boolean algebras, infinite combinatorics and large cardinals). 1979 edition.
Synopsis
An advanced-level treatment of the basics of set theory, this text offers students a firm foundation, stopping just short of the areas employing model-theoretic methods. Geared toward upper-level undergraduate and graduate students, it consists of two parts: the first covers pure set theory, including the basic motions, order and well-foundedness, cardinal numbers, the ordinals, and the axiom of choice and some of it consequences; the second deals with applications and advanced topics such as point set topology, real spaces, Boolean algebras, and infinite combinatorics and large cardinals. An appendix comprises useful information on eliminability and conservation theorems, and numerous exercises help students test their grasp of each topic. 1979 edition. 20 figures.
Table of Contents
Part A. Pure Set Theory
and#160; Chapter I. The Basic Notions
and#160;and#160;and#160; 1. The Basic Language of Set Theory
and#160;and#160;and#160; 2. The Axioms of Extensionality and Comprehension
and#160;and#160;and#160; 3. Classes, Why and How
and#160;and#160;and#160; 4. Classes, the formal Introduction
and#160;and#160;and#160; 5. The Axioms of Set Theory
and#160;and#160;and#160; 6. Relations and functions
and#160; Chapter II. Order and Well-Foundedness
and#160;and#160;and#160; 1. Order
and#160;and#160;and#160; 2. Well-Order
and#160;and#160;and#160; 3. Ordinals
and#160;and#160;and#160; 4. Natural Numbers and finite Sequences
and#160;and#160;and#160; 5. Well-Founded Relations
and#160;and#160;and#160; 6. Well-Founded Sets
and#160;and#160;and#160; 7. The Axiom of Foundation
and#160; Chapter III. Cardinal Numbers
and#160;and#160;and#160; 1. Finite Sets
and#160;and#160;and#160; 2. The Partial Order of the Cardinals
and#160;and#160;and#160; 3. The Finite Arithmetic of the Cardinals
and#160;and#160;and#160; 4. The Infinite Arithmetic of the Well Orderd Cardinals
and#160; Chapter IV. The Ordinals
and#160;and#160;and#160; 1. Ordinal Addition and Multiplication
and#160;and#160;and#160; 2. Ordinal Exponentiation
and#160;and#160;and#160; 3. Cofinality and Regular Ordinals
and#160;and#160;and#160; 4. Closed Unbounded Classes and Stationery Classes
and#160; Chapter V. The Axiom of Choice and Some of Its Consequences
and#160;and#160;and#160; 1. The Axiom of Choice and Equivalent Statements
and#160;and#160;and#160; 2. Some Weaker Versions of the Axiom of Choice
and#160;and#160;and#160; 3. Definable Sets
and#160;and#160;and#160; 4. Set Theory with Global Choice
and#160;and#160;and#160; 5. Cardinal Exponentiation
Part B. Applications and Advanced Topics
and#160; Chapter VI. A Review of Point Set Topology
and#160;and#160;and#160; 1. Basic concepts
and#160;and#160;and#160; 2. Useful Properties and Operations
and#160;and#160;and#160; 3. Category, Baire and Borel Sets
and#160; Chapter VII. The Real Spaces
and#160;and#160;and#160; 1. The Real Numbers
and#160;and#160;and#160; 2. The Separable Complete Metric Spaces
and#160;and#160;and#160; 3. The Close Relationship Between the Real Numbers, the Cantor Space and the Baire Space
and#160; Chapter VIII. Boolean Algebras
and#160;and#160;and#160; 1. The Basic Theory
and#160;and#160;and#160; 2. Prime Ideals and Representation
and#160;and#160;and#160; 3. Complete Boolean Algebras
and#160;and#160;and#160; 4. Martin's Axiom
and#160; Chapter IX. Infinite Combinatorics and Large Cardinals
and#160;and#160;and#160; 1. The Axiom of Constructibility
and#160;and#160;and#160; 2. Trees
and#160;and#160;and#160; 3. Partition Properties
and#160;and#160;and#160; 4. Measurable Cardinals
Appendix X. The Eliminability and Conservation Theorems
and#160; Bibliography; Additional Bibliography; Index of Notation; Index
Appendix Corrections and Additions