An Algebraic Introduction to Mathematical Logic | Date: 21 April 2011, 14:35
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Preface:
This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a substantial course on abstract algebra. Consequently, our treatment of the subject is algebraic. Although we assume a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of the exercises. We also assume a familiarity with the main ideas of set theory,
including cardinal numbers and Zorn's Lemma.
In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model of logic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-language. We are not attempting to construct a foundation on which all mathematics is to be based-rather, any conclusions to be drawn about the foundations of mathematics come only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory.
The construction of our model is greatly simplified by our using universal algebra in a way which enables us to dispense with the usual discussion of essentially notational questions about well-formed formulae. All questions and constructions relating to the set of well-formed formulae are handled by our Theorems 2.2 and 4.3 of Chapter 1. Our use of universal algebra also provides us with a convenient method for discussing free variables (and avoiding reference to bound variables), and it also permits a simple neat
statement of the Substitution Theorem (Theorems 4.11 of Chapter II and 4.3 of Chapter IV).
Chapter I develops the necessary amount of universal algebra. Chapters II and III respectively construct and analyse a model of the Propositional Calculus, introducing in simple form many of the ideas needed for the more complex First-Order Predicate Calculus, which is studied in Chapter IV. In Chapter V, we consider first-order mathematical theories, i.e., theories built on the First-Order Predicate Calculus, thus building models ofparts of mathematics. As set theory is usually regarded as the basis on which the rest of mathematics is constructed, we devote Chapter VI to a study of first-order Zermelo-Fraenkel Set Theory. Chapter VII, on Ultraproducts, discusses a
technique for constructing new models of a theory from a given collection ofmodels. Chapter VIII, which is an introduction to Non-Standard Analysis, is included as an example of mathematical logic assisting in the study of another branch ofmathematics. Decision processes are investigated in Chapter IX, and we prove there the non-existence ofdecision processes for a number of problems. In Chapter X, we discuss two decision problems from other branches of mathematics and indicate how the results of Chapter IX may be applied.
This book is intended to make mathematical logic 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 have included a number of exercises. Some ofthese fill in minor gaps in our exposition ofthe section in which they appear. Others indicate aspects ofthe subject which have been ignored in the text. Some are to help in understanding the text by applying ideas and methods to special cases. Occasionally, an exercise asks for the construction of a FORTRAN program. In such cases, the solution should be based on integer arithmetic, and not depend on any special logical properties of FORTRAN or of any other programming language.
The layout of the text is as follows. Each chapter is divided into numbered sections, and definitions, theorems, exercises, etc. are numbered consecutively within each section. For example, the number 2.4 refers to the fourth item in the second section of the current chapter. A reference to an item in some other chapter always includes the chapter number in addition to item and section numbers.
We thank the many mathematical colleagues, particularly Paul Halmos and Peter Hilton, who encouraged and advised us in this project. We are
especially indebted to Gordon Monro for suggesting many improvements and for providing many exercises. We thank Mrs. Blakestone and Miss
Kicinski for the excellent typescript they produced.
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