An Algebraic Introduction to Mathematical Logic

✍ Scribed by Donald W. Barnes and John M. Mack

• Publisher: Springer
• Year: 1975
• Tongue: English
• Leaves: 129
• Series: Graduate Texts in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This book is intended for mathematicians. Our treatment of mathematical logic is algebraic. Although we assurne a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. This book is intended to make mathematica llogic available to mathematicians working in other branches of mathematics. We have included what we consider to be the essential basic theory, some useful techniques, and some indications of ways in which the theory might be of use in other branches of mathematics. We also have included a number of exercises.

✦ Table of Contents

Cover
Title
Copyright
Preface
Table of Contents
Chapter I Universal Algebra
§1 Introduction
§2 Free Algebras
§3 Varieties of Algebras
§4 Relatively Free Algebras
Chapter II Propositional Calculus
§1 Introduction
§2 Algebras of Propositions
§3 Truth in the Propositional Calculus
§4 Proof in the Propositional Calculus
Chapter III Properties of the Propositional Calculus
§1 Introduction
§2 Soundness and Adequacy of Prop(X)
§3 Truth Functions and Decidability for Prop(X)
Chapter IV Predicate Calculus
§1 Algebras of Predicates
§2 Interpretations
§3 Proof in Pred(V, R)
§4 Properties of Pred(V, R)
Chapter V First-Order Mathematics
§1 Predicate Calculus with Identity
§2 First-Order Mathematical Theories
§3 Properties of First-Order Theories
§4 Reduction of Quantifiers
Chapter VI Zermelo-Fraenkel Set Theory
§1 Introduction
§2 The Axioms of ZF
§3 First-Order ZF
§4 The Peano Axioms
Chapter VII Ultraproducts
§1 Ultraproducts
§2 Non-Principal Ultrafilters
§3 The Existence of an Aigebraic Closure
§4 Non-Trivial Ultrapowers
§5 Ultrapowers of Number Systems
§6 Direct Limits
Chapter VIII Non-Standard Models
§1 Elementary Standard Systems
§2 Reduction of the Order
§3 Enlargements
§4 Standard Relations
§5 Internal Relations
§6 Non-Standard Analysis
Chapter IX Turing Machines and Gödel Numbers
§1 Decision Processes
§2 Turing Machines
§3 Recursive Functions
§4 Gödel Numbers
§5 Insoluble Problems in Mathematics
§6 Insoluble Problems in Arithmetic
§7 Undecidability of the Predicate Calculus
Chapter X Hilbert's Tenth Problem, Word Problems
§1 Hilbert's Tenth Problem
§2 Word Problems
References and Further Reading
Index of Notations
Subject Index