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- Preface
1: The Axiom of Extension
2: The Axiom of Specification
3: Unordered Pairs
4: Unions and Intersections
5: Complements and Powers
6: Ordered Pairs
7: Relations
8: Functions
9: Families
10: Inverses and Composites
11: Numbers
12: The Peano Axioms
13: Arithmetic
14: Order
15: The Axiom of Choice
16: Zorn's Lemma
17: Well Ordering
18: Transfinite Recursion
19: Ordinal Numbers
2: Sets of Ordinal Numbers
21: Ordinal Arithmetic
22: The Schr der-Bernstein Theorem
23: Countable Sets
24: Cardinal Arithmetic
25: Carnidal numbers.
Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdomis still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.
1: The Axiom of Extension
2: The Axiom of Specification
3: Unordered Pairs
4: Unions and Intersections
5: Complements and Powers
6: Ordered Pairs
7: Relations
8: Functions
9: Families
10: Inverses and Composites
11: Numbers
12: The Peano Axioms
13: Arithmetic
14: Order
15: The Axiom of Choice
16: Zorn's Lemma
17: Well Ordering
18: Transfinite Recursion
19: Ordinal Numbers
2: Sets of Ordinal Numbers
21: Ordinal Arithmetic
22: The Schr der-Bernstein Theorem
23: Countable Sets
24: Cardinal Arithmetic
25: Carnidal numbers.
Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdomis still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.
From the reviews:
"This book is a very specialized but broadly useful introduction to set theory. It is aimed at 'the beginning student of advanced mathematics' ... who wants to understand the set-theoretic underpinnings of the mathematics he already knows or will learn soon. It is also useful to the professional mathematician who knew these underpinnings at one time but has now forgotten exactly how they go. ... A good reference for how set theory is used in other parts of mathematics ... ." (Allen Stenger, The Mathematical Association of America, September, 2011)
"This book is a very specialized but broadly useful introduction to set theory. It is aimed at 'the beginning student of advanced mathematics' ... who wants to understand the set-theoretic underpinnings of the mathematics he already knows or will learn soon. It is also useful to the professional mathematician who knew these underpinnings at one time but has now forgotten exactly how they go. ... A good reference for how set theory is used in other parts of mathematics ... ." (Allen Stenger, The Mathematical Association of America, September, 2011)