Introduction to Mathematical Logic [antikvár]

In this book we have attempted to present a compact introduction to some of the principal topics of mathematical logic. In order to give a full and precise treatment of the more important basic subjects, certain subsidiary topics, such as modal, combinatory, and intuitionistic logics, and some interesting advanced topics, such as degrees of recursive un-solvability, have had to be omitted. In the belief that beginners should be exposed to the most natural and easiest proofs, free-swinging set-theoretic methods have been used. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. After all, if we are to be expelled from "Cantor's paradise" (as non-constructive set theory was called by Hilbert), at least we should know what we are missing. The five chapters of the book can be covered in two semesters, but, for a one-semester course. Chapters 1 through 3 will be quite adequate (omitting, if hurried. Sections 5 and 6 of Chapter 1 and Sections 10, 11, and 12 of Chapter 2). The convention has been adopted of prefixing a superscript "D" to any section or exercise which will probably be difiScult for a beginner, and a superscript "A" to any section or exercise which presupposes familiarity with a topic that has not been carefully explained in the text. Bibliographical references are given to the best source of information, which is not always the earliest paper; hence these references give no indication as to priority. For example, Boone [1959] gives the most complete account of his work on the word problem, which was actually done independently of and about the same time as Novikov's work [1955]. The present book is an expansion of lecture notes for a one-semester course in mathematical logic given by the author at Columbia University from 1958 to 1960 and at Queens College in 1961 and 1962. The author hopes that it can be read with ease by anyone with a certain amount of experience in abstract mathematical thought, but there is no specific prerequisite. The author would like to thank J. Barkley Rosser for encouragement and guidance during his graduate studies in logic, and he would like to acknowledge also the obvious debt owed to the books of Hilbert-Bernays, 1934, 1939; Kleene, 1952; Rosser, 1953; and Church, 1956. Elliott Mendblson Queens, New York January 1968