A concise and essential resource for basic logic concepts, methods and information.
The book is an excellent resource for the working mathematical scientist. Graduate students, practitioners and professionals in computer science and engineering, or the systems scientist who needs a quick sketch of a key idea from logic, will find it in this self-contained, accessible, and easy-to-use reference.
Rezensionen / Stimmen
"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
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Professional/practitioner
Produkt-Hinweis
Fadenheftung
Gewebe-Einband
Illustrationen
Maße
Höhe: 241 mm
Breite: 159 mm
Dicke: 18 mm
Gewicht
ISBN-13
978-0-8176-4220-4 (9780817642204)
DOI
10.1007/978-1-4612-0115-1
Schweitzer Klassifikation
Krantz, S.G., Washington University, St. Louis, USA
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.
