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Cover: 9789401021142 | An Outline of Mathematical Logic | Andrzej Grzegorczyk | Taschenbuch
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An Outline of Mathematical Logic
Fundamental Results and Notions Explained with all Details
Taschenbuch von Andrzej Grzegorczyk
Sprache: Englisch

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Beschreibung
Recent years have seen the appearance of many English-language hand­ books of logic and numerous monographs on topical discoveries in the foundations of mathematics. These publications on the foundations of mathematics as a whole are rather difficult for the beginners or refer the reader to other handbooks and various piecemeal contribu­ tions and also sometimes to largely conceived "mathematical fol­ klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of the now classical results in the foundations of mathematics. Hence the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of such words as evident and obvious in proofs to a minimum. This is why the book, it is believed, may be helpful in teaching or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subject, which does not enrich the essential knowledge in founda­ tions but can discourage the beginner by their abstract form. A. G.
Recent years have seen the appearance of many English-language hand­ books of logic and numerous monographs on topical discoveries in the foundations of mathematics. These publications on the foundations of mathematics as a whole are rather difficult for the beginners or refer the reader to other handbooks and various piecemeal contribu­ tions and also sometimes to largely conceived "mathematical fol­ klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of the now classical results in the foundations of mathematics. Hence the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of such words as evident and obvious in proofs to a minimum. This is why the book, it is believed, may be helpful in teaching or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subject, which does not enrich the essential knowledge in founda­ tions but can discourage the beginner by their abstract form. A. G.
Inhaltsverzeichnis
to the Problems of the Foundations of Mathematics.- 1. Mathematical Domains.- 2. Examples of Mathematical Domains.- 3. Selected Kinds of Relations and Functions.- 4. Logical Analysis of Mathematical Concepts.- 5. Zermelo's Set Theory.- 6. Set-Theoretical Approach to Relations and Functions.- 7. The Genetic Construction of Natural Numbers.- 8. Expansion of the Concept of Number.- 9. Construction of New Mathematical Domains.- 10. Subdomains, Homomorphisms, Isomorphisms.- 11. Products. Real Numbers.- I. The Classical Logical Calculus.- 1. The Classical Characteristics of the Sentential Connectives.- 2. Tautologies in the Classical Sentential Calculus and Their Applications to Certain Mathematical Considerations.- 3. An Axiomatic Approach to the Sentential Calculus.- 4. The Classical Concept of Quantifier.- 5. The Predicate Calculus in the Traditional Interpretation.- 6. Reduction of Quantifier Rules to Axioms, c.l.c Tautologies True in the Empty Domain.- 7. The Concepts of Consequence and Theory. Applications of the Logical Calculus to the Formalization of Mathematical Theories.- 8. The Logical Functional Calculus L* and Its Applications to the Formalization of Theories with Functions.- 9. Certain Syntactic Properties of the Classical Logical Calculus.- 10. On Definitions.- II. Models of Axiomatic Theories.- 1. The Concept of Satisfaction.- 2. The Concepts of Truth and Model. The Properties of the Set of Sentences True in a Model.- 3. Existence of co-complete Extensions and Denumerable Models.- 4. Some Other Concepts and Results in Model Theory.- 5. Skolem's Elimination of Quantifiers, Consistency of Compound Theories and Interpolation Theorems.- 6. Definability.- III. Logical Hierarchy of Concepts.- 1. The Concept of Effectiveness in Arithmetic.- 2. Some Properties ofComputable Functions 417.- 3. Effectiveness of Methods of Proof.- 4. Representability of Computable Relations in Arithmetic.- 5. Problems of Decidability.- 6. Logical Hierarchy of Arithmetic Concepts.- Supplement. a Historical Outline.- Index of Symbols.- Index of Names.
Details
Erscheinungsjahr: 2011
Fachbereich: Allgemeines
Genre: Philosophie
Jahrhundert: Antike
Rubrik: Geisteswissenschaften
Thema: Lexika
Medium: Taschenbuch
Reihe: Synthese Library
Inhalt: 606 S.
ISBN-13: 9789401021142
ISBN-10: 9401021147
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Grzegorczyk, Andrzej
Übersetzung: Zawadowski, Waclaw
Wojtasiewicz, Olgierd
Auflage: Softcover reprint of the original 1st ed. 1974
Hersteller: Springer Netherland
Springer Netherlands
Synthese Library
Maße: 229 x 152 x 33 mm
Von/Mit: Andrzej Grzegorczyk
Erscheinungsdatum: 25.11.2011
Gewicht: 0,873 kg
Artikel-ID: 105625507
Inhaltsverzeichnis
to the Problems of the Foundations of Mathematics.- 1. Mathematical Domains.- 2. Examples of Mathematical Domains.- 3. Selected Kinds of Relations and Functions.- 4. Logical Analysis of Mathematical Concepts.- 5. Zermelo's Set Theory.- 6. Set-Theoretical Approach to Relations and Functions.- 7. The Genetic Construction of Natural Numbers.- 8. Expansion of the Concept of Number.- 9. Construction of New Mathematical Domains.- 10. Subdomains, Homomorphisms, Isomorphisms.- 11. Products. Real Numbers.- I. The Classical Logical Calculus.- 1. The Classical Characteristics of the Sentential Connectives.- 2. Tautologies in the Classical Sentential Calculus and Their Applications to Certain Mathematical Considerations.- 3. An Axiomatic Approach to the Sentential Calculus.- 4. The Classical Concept of Quantifier.- 5. The Predicate Calculus in the Traditional Interpretation.- 6. Reduction of Quantifier Rules to Axioms, c.l.c Tautologies True in the Empty Domain.- 7. The Concepts of Consequence and Theory. Applications of the Logical Calculus to the Formalization of Mathematical Theories.- 8. The Logical Functional Calculus L* and Its Applications to the Formalization of Theories with Functions.- 9. Certain Syntactic Properties of the Classical Logical Calculus.- 10. On Definitions.- II. Models of Axiomatic Theories.- 1. The Concept of Satisfaction.- 2. The Concepts of Truth and Model. The Properties of the Set of Sentences True in a Model.- 3. Existence of co-complete Extensions and Denumerable Models.- 4. Some Other Concepts and Results in Model Theory.- 5. Skolem's Elimination of Quantifiers, Consistency of Compound Theories and Interpolation Theorems.- 6. Definability.- III. Logical Hierarchy of Concepts.- 1. The Concept of Effectiveness in Arithmetic.- 2. Some Properties ofComputable Functions 417.- 3. Effectiveness of Methods of Proof.- 4. Representability of Computable Relations in Arithmetic.- 5. Problems of Decidability.- 6. Logical Hierarchy of Arithmetic Concepts.- Supplement. a Historical Outline.- Index of Symbols.- Index of Names.
Details
Erscheinungsjahr: 2011
Fachbereich: Allgemeines
Genre: Philosophie
Jahrhundert: Antike
Rubrik: Geisteswissenschaften
Thema: Lexika
Medium: Taschenbuch
Reihe: Synthese Library
Inhalt: 606 S.
ISBN-13: 9789401021142
ISBN-10: 9401021147
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Grzegorczyk, Andrzej
Übersetzung: Zawadowski, Waclaw
Wojtasiewicz, Olgierd
Auflage: Softcover reprint of the original 1st ed. 1974
Hersteller: Springer Netherland
Springer Netherlands
Synthese Library
Maße: 229 x 152 x 33 mm
Von/Mit: Andrzej Grzegorczyk
Erscheinungsdatum: 25.11.2011
Gewicht: 0,873 kg
Artikel-ID: 105625507
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