Logic plays a central conceptual role in modern mathematics. However, mathematical logic has grown into one of the most recondite areas of mathematics. As a result, most of modern logic is inaccessible to all but the specialist. This new book is a resource that provides a quick introduction and review of the key topics in logic for the computer scientist, engineer, or mathematician. Handbook of Logic and Proof Techniques for Computer Science presents the elements of modern logic, including many current topics, to the reader having only basic mathematical literacy. Computer scientists will find…mehr
Logic plays a central conceptual role in modern mathematics. However, mathematical logic has grown into one of the most recondite areas of mathematics. As a result, most of modern logic is inaccessible to all but the specialist. This new book is a resource that provides a quick introduction and review of the key topics in logic for the computer scientist, engineer, or mathematician. Handbook of Logic and Proof Techniques for Computer Science presents the elements of modern logic, including many current topics, to the reader having only basic mathematical literacy. Computer scientists will find specific examples and important ideas such as axiomatics, recursion theory, decidability, independence, completeness, consistency, model theory, and P/NP completeness. The book contains definitions, examples and discussion of all of the key ideas in basic logic, but also makes a special effort to cut through the mathematical formalism, difficult notation, and esoteric terminology that is typical of modern mathematical logic. T This handbook delivers cogent and self-contained introductions to critical advanced topics, including: Godel`s completeness and incompleteness theorems Methods of proof, cardinal and ordinal numbers, the continuum hypothesis, the axiom of choice, model theory, and number systems and their construction Extensive treatment of complexity theory and programming applications Applications to algorithms in Boolean algebra Discussion of set theory and applications of logic The book is an excellent resource for the working mathematical scientist. The graduate student or professional in computer science and engineering or the systems scientist who needs to have a quick sketch of a key idea from logic will find it here in this self-contained, accessible, and easy-to-use reference.
Steven Krantz, Ph.D., is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 45 books on mathematics, including Calculus Demystified, another popular title in this series. He lives in St. Louis, Missouri.
Inhaltsangabe
Preface Notation and First-Order Logic Semantics and Syntax Axiomatics and Formalism in Mathematics The Axioms of Set Theory Elementary Set Theory Recursive Functions The Number Systems Methods of Mathematical Proof The Axiom of Choice Proof Theory Category Theory Complexity Theory Boolean Algebra The Word Problem List of Notation and Logic Glossary Terms from Mathematical and Sentential Logic A Guide to the Literature Bibliography Index
1 Notation and First-Order Logic.- 1.1 The Use of Connectives.- 1.2 Truth Values and Truth Tables.- 1.3 The Use of Quantifiers.- 1.4 Gödel's Completeness Theorem.- 1.5 Second-Order Logic.- 2 Semantics and Syntax.- 2.1 Elementary Symbols.- 2.2 Well-Formed Formulas or wffs [Syntax].- 2.3 Free and Bound Variables (Syntax).- 2.4 The Semantics of First-Order Logic.- 3 Axiomatics and Formalism in Mathematics.- 3.1 Basic Elements.- 3.2 Models.- 3.3 Consistency.- 3.4 Gödel's Incompleteness Theorem.- 3.5 Decidability and Undecidability.- 3.6 Independence.- 4 The Axioms of Set Theory.- 4.1 Introduction.- 4.2 Axioms and Discussion.- 4.3 Concluding Remarks.- 5 Elementary Set Theory.- 5.1 Set Notation.- 5.2 Sets, Subsets, and Elements.- 5.3 Binary Operations on Sets.- 5.4 Relations and Equivalence Relation.- 5.5 Equivalence Relations.- 5.6 Number Systems.- 5.7 Functions.- 5.8 Cardinal Numbers.- 5.9 A Word About Classes.- 5.10 Fuzzy Set Theory.- 5.11 The Lambda Calculus.- 5.12 Sequences.- 5.13 Bags.- 6 Recursive Functions.- 6.1 Introductory Remarks.- 6.2 Primitive Recursive Functions.- 6.3 General Recursive Functions.- 7 The Number Systems.- 7.1 The Natural Numbers.- 7.2 The Integers.- 7.3 The Rational Numbers.- 7.4 The Real Numbers.- 7.5 The Complex Numbers.- 7.6 The Quaternions.- 7.7 The Cayley Numbers.- 7.8 Nonstandard Analysis.- 8 Methods of Mathematical Proof.- 8.1 Axiomatics.- 8.2 Proof by Induction.- 8.3 Proof by Contradiction.- 8.4 Direct Proof.- 8.5 Other Methods of Proof.- 9 The Axiom of Choice.- 9.1 Enunciation of the Axiom.- 9.2 Examples of the Use of the Axiom of Choice.- 9.3 Consequences of the Axiom of Choice.- 9.4 Paradoxes.- 9.5 The Countable Axiom of Choice.- 9.6 Consistency of the Axiom of Choice.- 9.7 Independence of the Axiom of Choice.- 10 Proof Theory.-10.1 General Remarks.- 10.2 Cut Elimination.- 10.3 Propositional Resolution.- 10.4 Interpolation.- 10.5 Finite Type.- 10.6 Beth's Definability Theorem.- 11 Category Theory.- 11.1 Introductory Remarks.- 11.2 Metacategories and Categories.- 12 Complexity Theory.- 12.1 Preliminary Remarks.- 12.2 Polynomial Complexity.- 12.3 Exponential Complexity.- 12.4 Two Tables for Complexity Theory.- 12.5 Problems of Class P.- 12.6 Problems of Class NP.- 12.7 NP-Completeness.- 12.8 Cook's Theorem.- 12.9 Examples of NP-Complete Problems.- 12.10 More on P/NP.- 12.11 Descriptive Complexity Theory.- 13 Boolean Algebra.- 13.1 Description of Boolean Algebra.- 13.2 Axioms of Boolean Algebra.- 13.3 Theorems in Boolean Algebra.- 13.4 Illustration of the Use of Boolean Logic.- 14 The Word Problem.- 14.1 Introductory Remarks.- 14.2 What Is a Group?.- 14.3 What Is a Free Group?.- 14.4 The Word Problem.- 14.5 Relations and Generators.- 14.6 Amalgams.- 14.7 Description of the Word Problem.- List of Notation from Logic.- Glossary of Terms from Mathematical and Sentential Logic.- A Guide to the Literature.
Preface Notation and First-Order Logic Semantics and Syntax Axiomatics and Formalism in Mathematics The Axioms of Set Theory Elementary Set Theory Recursive Functions The Number Systems Methods of Mathematical Proof The Axiom of Choice Proof Theory Category Theory Complexity Theory Boolean Algebra The Word Problem List of Notation and Logic Glossary Terms from Mathematical and Sentential Logic A Guide to the Literature Bibliography Index
1 Notation and First-Order Logic.- 1.1 The Use of Connectives.- 1.2 Truth Values and Truth Tables.- 1.3 The Use of Quantifiers.- 1.4 Gödel's Completeness Theorem.- 1.5 Second-Order Logic.- 2 Semantics and Syntax.- 2.1 Elementary Symbols.- 2.2 Well-Formed Formulas or wffs [Syntax].- 2.3 Free and Bound Variables (Syntax).- 2.4 The Semantics of First-Order Logic.- 3 Axiomatics and Formalism in Mathematics.- 3.1 Basic Elements.- 3.2 Models.- 3.3 Consistency.- 3.4 Gödel's Incompleteness Theorem.- 3.5 Decidability and Undecidability.- 3.6 Independence.- 4 The Axioms of Set Theory.- 4.1 Introduction.- 4.2 Axioms and Discussion.- 4.3 Concluding Remarks.- 5 Elementary Set Theory.- 5.1 Set Notation.- 5.2 Sets, Subsets, and Elements.- 5.3 Binary Operations on Sets.- 5.4 Relations and Equivalence Relation.- 5.5 Equivalence Relations.- 5.6 Number Systems.- 5.7 Functions.- 5.8 Cardinal Numbers.- 5.9 A Word About Classes.- 5.10 Fuzzy Set Theory.- 5.11 The Lambda Calculus.- 5.12 Sequences.- 5.13 Bags.- 6 Recursive Functions.- 6.1 Introductory Remarks.- 6.2 Primitive Recursive Functions.- 6.3 General Recursive Functions.- 7 The Number Systems.- 7.1 The Natural Numbers.- 7.2 The Integers.- 7.3 The Rational Numbers.- 7.4 The Real Numbers.- 7.5 The Complex Numbers.- 7.6 The Quaternions.- 7.7 The Cayley Numbers.- 7.8 Nonstandard Analysis.- 8 Methods of Mathematical Proof.- 8.1 Axiomatics.- 8.2 Proof by Induction.- 8.3 Proof by Contradiction.- 8.4 Direct Proof.- 8.5 Other Methods of Proof.- 9 The Axiom of Choice.- 9.1 Enunciation of the Axiom.- 9.2 Examples of the Use of the Axiom of Choice.- 9.3 Consequences of the Axiom of Choice.- 9.4 Paradoxes.- 9.5 The Countable Axiom of Choice.- 9.6 Consistency of the Axiom of Choice.- 9.7 Independence of the Axiom of Choice.- 10 Proof Theory.-10.1 General Remarks.- 10.2 Cut Elimination.- 10.3 Propositional Resolution.- 10.4 Interpolation.- 10.5 Finite Type.- 10.6 Beth's Definability Theorem.- 11 Category Theory.- 11.1 Introductory Remarks.- 11.2 Metacategories and Categories.- 12 Complexity Theory.- 12.1 Preliminary Remarks.- 12.2 Polynomial Complexity.- 12.3 Exponential Complexity.- 12.4 Two Tables for Complexity Theory.- 12.5 Problems of Class P.- 12.6 Problems of Class NP.- 12.7 NP-Completeness.- 12.8 Cook's Theorem.- 12.9 Examples of NP-Complete Problems.- 12.10 More on P/NP.- 12.11 Descriptive Complexity Theory.- 13 Boolean Algebra.- 13.1 Description of Boolean Algebra.- 13.2 Axioms of Boolean Algebra.- 13.3 Theorems in Boolean Algebra.- 13.4 Illustration of the Use of Boolean Logic.- 14 The Word Problem.- 14.1 Introductory Remarks.- 14.2 What Is a Group?.- 14.3 What Is a Free Group?.- 14.4 The Word Problem.- 14.5 Relations and Generators.- 14.6 Amalgams.- 14.7 Description of the Word Problem.- List of Notation from Logic.- Glossary of Terms from Mathematical and Sentential Logic.- A Guide to the Literature.
Rezensionen
"This is really what it promises to be-a good handbook: supple, self-contained, providing the necessary and sufficient working resources . . . it is more than [one] expect[s]: the rigor of usefulness and conciseness exceeds or equals . . . the pleasure of reading it."
-Zentralblatt Math
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