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handbooks and various piecemeal contribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results.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.
Author(s): Andrzej Grzegorczyk
Edition: Softcover reprint of the original 1st ed. 1974
Publisher: Springer
Year: 2011
Language: English
Pages: 606
Preface
Contents
Introduction 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
Chapter 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 Oassical 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
Chapter 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 ω-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
Chapter III. Logical hierarchy of concepts
1. The Concept of Effectiveness in Arithmetic
2. Some Properties of Computable 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
Bibliography
Index of symbols
Index of names
Subject index
