Classical Mathematical Logic: The Semantic Foundations of Logic

✍ Scribed by Richard L. Epstein; Leslaw W. Szczerba

• Publisher: Princeton University Press
• Year: 2011
• Tongue: English
• Leaves: 544
• Edition: Course Book
• Category: Library

✦ Synopsis

In Classical Mathematical Logic, Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations.

The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference.

Classical Mathematical Logic presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.

✦ Table of Contents

Contents
Preface
Acknowledgments
Introduction
I. Classical Propositional Logic
II. Abstracting and Axiomatizing Classical Propositional Logic
III. The Language of Predicate Logic
IV. The Semantics of Classical Predicate Logic
V. Substitutions and Equivalences
VI. Equality
VII. Examples of Formalization
VIII. Functions
IX. The Abstraction of Models
X. Axiomatizing Classical Predicate Logic
XI. The Number of Objects in the Universe of a Model
XII. Formalizing Group Theory
XIII. Linear Orderings
XIV. Second-Order Classical Predicate Logic
XV. The Natural Numbers
XVI. The Integers and Rationals
XVII. The Real Numbers
XVIII. One-Dimensional Geometry
XIX. Two-Dimensional Euclidean Geometry
XX. Translations within Classical Predicate Logic
XXI. Classical Predicate Logic with Non-Referring Names
XXII. The Liar Paradox
XXIII. On Mathematical Logic and Mathematics
Appendix: The Completeness of Classical Predicate Logic Proved by Gödel’s Method
Summary of Formal Systems
Bibliography
Index of Notation
Index