Synopses & Reviews
Set Theory and Logic
is the result of a course of lectures for advanced undergraduates, developed at Oberlin College for the purpose of introducing students to the conceptual foundations of mathematics. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. One of the most complex and essential of modern mathematical innovations, the theory of sets (crucial to quantum mechanics and other sciences), is introduced in a most careful concept manner, aiming for the maximum in clarity and stimulation for further study in set logic.
Contents include: Sets and Relations — Cantor's concept of a set, etc.
Natural Number Sequence — Zorn's Lemma, etc.
Extension of Natural Numbers to Real Numbers
Logic — the Statement and Predicate Calculus, etc.
Informal Axiomatic Mathematics
Boolean AlgebraInformal Axiomatic Set TheorySeveral Algebraic Theories — Rings, Integral Domains, Fields, etc.
First-Order Theories — Metamathematics, etc.
Symbolic logic does not figure significantly until the final chapter. The main theme of the book is mathematics as a system seen through the elaboration of real numbers; set theory and logic are seen s efficient tools in constructing axioms necessary to the system.
Mathematics students at the undergraduate level, and those who seek a rigorous but not unnecessarily technical introduction to mathematical concepts, will welcome the return to print of this most lucid work.
"Professor Stoll . . . has given us one of the best introductory texts we have seen." — Cosmos.
"In the reviewer's opinion, this is an excellent book, and in addition to its use as a textbook (it contains a wealth of exercises and examples) can be recommended to all who wish an introduction to mathematical logic less technical than standard treatises (to which it can also serve as preliminary reading)." — Mathematical Reviews.
Explores sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories.
"The best introductory text we have seen." — Cosmos. Lucidly and gradually explains sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories. Its clarity makes this book excellent for self-study.
Bibliography: p. 457-464.
Table of Contents
Chapter 1 SETS AND RELATIONS
1. Cantor's Concept of a Set
2. The Basis of Intuitive Set Theory
4. Operations for Sets
5. The Algebra of Sets
7. Equivalence Relations
9. Composition and Inversion for Functions
10. Operations for Collections of Sets
11. Ordering Relations
Chapter 2 THE NATURAL NUMBER SEQUENCE AND ITS GENERALIZATIONS
1. The Natural Number Sequence
2. Proof and Definition by Induction
3. Cardinal Numbers
4. Countable Sets
5. Cardinal Arithmetic
6. Order Types
7. Well-ordered Sets and Ordinal Numbers
8. "The Axiom of Choice, the Well-ordering Theorem, and Zorn's Lemma"
9. Further Properties of Cardinal Numbers
10. Some Theorems Equivalent to the Axiom of Choice
11. The Paradoxes of Intuitive Set Theory
Chapter 3 THE EXTENSION OF THE NATURAL NUMBERS TO THE REAL NUMBERS
1. The System of Natural Numbers
4. Rational Numbers
5. Cauchy Sequences of Rational Numbers
6. Real Numbers
7. Further Properties of the Real Number System
Chapter 4 LOGIC
1. The Statement Calculus. Sentential Connectives
2. The Statement Calculus. Truth Tables
3. The Statement Calculus. Validity
4. The Statement Calculus. Consequence
5. The Statement Calculus. Applications
6. The Predicate Calculus. Symbolizing Everyday Language
7. The Predicate Calculus. A Formulation
8. The Predicate Calculus. Validity
9. The Predicate Calculus. Consequence
Chapter 5 INFORMAL AXIOMATIC MATHEMATICS
1. The Concept of an Axiomatic Theory
2. Informal Theories
3. Definitions of Axiomatic Theories by Set-theoretical Predicates
4. Further Features of Informal Theories
Chapter 6 BOOLEAN ALGEBRAS
1. A Definition of a Boolean Algebra
2. Some Basic Properties of a Boolean Algebra
3. Another Formulation of the Theory
4. Congruence Relations for a Boolean Algebra
5. Representations of Boolean Algebras
6. Statement Calculi as Boolean Algebras
7. Free Boolean Algebras
8. Applications of the Theory of Boolean Algebras to Statement Calculi
9. Further Interconnections between Boolean Algebras and Statement Calculi
Chapter 7 INFORMAL AXIOMATIC SET THEORY
1. The Axioms of Extension and Set Formation
2. The Axiom of Pairing
3. The Axioms of Union and Power Set
4. The Axiom of Infinity
5. The Axiom of Choice
6. The Axiom Schemas of Replacement and Restriction
7. Ordinal Numbers
8. Ordinal Arithmetic
9. Cardinal Numbers and Their Arithmetic
10. The von Neumann-Bernays-Gödel Theory of Sets
Chapter 8 SEVERAL ALGEBRAIC THEORIES
1. Features of Algebraic Theories
2. Definition of a Semigroup
3. Definition of a Group
5. Coset Decompositions and Congruence Relations for Groups
6. "Rings, Integral Domains, and Fields"
7. Subrings and Difference Rings
8. A Characterization of the System of Integers
9. A Characterization of the System of Rational Numbers
10. A Characterization of the Real Number System
Chapter 9 FIRST-ORDER THEORIES
1. Formal Axiomatic Theories
2. The Statement Calculus as a Formal Axiomatic Theory
3. Predicate Calculi of First Order as Formal Axiomatic Theories
4. First-order Axiomatic Theories
6. Consistency and Satisfiability of Sets of Formulas
7. "Consistency, Completeness, and Categoricity of First-order Theories"
8. Turing Machines and Recursive Functions
9. Some Undecidable and Some Decidable Theories
10. Gödel's Theorems
11. Some Further Remarks about Set Theory
Symbols and Notation