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

by Andrzej Grzegorczyk

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 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.- Index of Symbols.- Index of Names.

608 pages, Paperback

First published September 30, 1974