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## Set theory and logic / by Robert R. Stoll.

Editor: San Francisco : W. H. Freeman, c1961, c1963Descripción: xiv, 474 p. ; 24 cmOtro título: Introduction to set theory and logic [Título de cubierta]Otra clasificación: 03-01
Contenidos:
```Chapter 1 SETS AND RELATIONS 
1. Cantor’s Concept of a Set 
2. The Basis of Intuitive Set Theory 
3. Inclusion 
4. Operations for Sets 
5. The Algebra of Sets 
6. Relations 
7. Equivalence Relations 
8. Functions 
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 
2. Differences 
3. Integers 
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 
1. 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 
4. Subgroups 
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 
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 
5. Metamathematics 
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 
References 
Symbols and Notation 
Author Index 
Subject Index 
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Chapter 1 SETS AND RELATIONS  --
1. Cantor’s Concept of a Set  --
2. The Basis of Intuitive Set Theory  --
3. Inclusion  --
4. Operations for Sets  --
5. The Algebra of Sets  --
6. Relations  --
7. Equivalence Relations  --
8. Functions  --
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  --
2. Differences  --
3. Integers  --
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  --
1. 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  --
4. Subgroups  --
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  --
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  --
5. Metamathematics  --
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  --
References  --
Symbols and Notation  --
Author Index  --
Subject Index  --

MR, 30 #11

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