An Outline of Mathematical Logic: Fundamental Results and Notions Explained with All Details

by A. Grzegorczyk

Overview

Recent years have seen the appearance of many English-Ianguage handbooks of logie and numerous monographs on topieal discoveries in the foundations of mathematies. These publications on the foundations of mathematies as a whole are rather difficult for the beginners or refer the reader to other handbooks and various pieeemeal eontribu­ tions and also sometimes to largely conceived "mathematical folklore" of unpublished results.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. Henee 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 sueh words as evident and obvious in proofs to aminimum. This is why the book, it is believed, may be helpful in teaehing 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 subjeet, which does not enrich the essential knowledge in foundations but ean discourage the beginner by their abstract form. A. G.

Product Details

ISBN-13:	9789027704474
Publisher:	Springer Netherlands
Publication date:	09/01/1974
Series:	Population and Community Biology Series
Edition description:	Softcover reprint of the original 1st ed. 1974
Pages:	596
Product dimensions:	5.98(w) x 9.02(h) x 0.05(d)

Table of Contents

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.- 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.