Handbook of Mathematical Logic (Volume 90) (Studies in Logic and the Foundations of Mathematics, Volume 90)

The handbook is divided into four parts: model theory, set theory, recursion theory and proof theory. Each of the four parts begins with a short guide to the chapters that follow. Each chapter is written for non-specialists in the field in question. Mathematicians will find that this book provides them with a unique opportunity to apprise themselves of developments in areas other than their own. Front Cover......Page 1 Handbook of Mathematical Logic......Page 4 Copyright Page......Page 5 Table of Contents......Page 11 Foreword......Page 8 Contributors......Page 9 Part A: Model Theory......Page 14 Guide to Part A......Page 16 A.1. An introduction to first-order logic......Page 18 A.2. Fundamentals of model theory......Page 60 A.3. Ultraproducts for algebraists......Page 118 A.4. Model completeness......Page 152 A.5. Homogenous sets......Page 194 A.6. Infinitesimal analysis of curves and surfaces......Page 210 A.7. Admissible sets and infinitary logic......Page 246 A.8. Doctrines in categorical logic......Page 296 Part B: Set Theory......Page 328 Guide to Part B......Page 330 B.1. Axioms of set theory......Page 334 B.2. About the axiom of choice......Page 358 B.3. Combinatorics......Page 384 B.4. Forcing, John......Page 416 B.5. Constructibility......Page 466 B.6. Martin’s Axiom......Page 504 B.7. Consistency results in topology......Page 516 Part C: Recursion Theory......Page 536 Guide to Part C......Page 538 C.1. Elements of recursion theory......Page 540 C.2. Unsolvable problems......Page 580 C.3. Decidable theories......Page 608 C.4. Degrees of unsolvability: a survey of results......Page 644 C.5. α -recursion theory......Page 666 C.6. Recursion in higher types......Page 694 C.7. An introduction to inductive definitions......Page 752 C.8. Descriptive set theory: Projective sets......Page 796 Part D: Proof Theory And Constructive Mathematics Guide To Part D......Page 830 Guide to Part D......Page 832 D.1. The incompleteness theorems......Page 834 D.2. Proof theory: Some applications of cut-elimination......Page 880 D.3. Herbrand’s Theorem and Gentzen’s notion of a direct proof......Page 910 D.4. Theories of finite type related to mathematical practice......Page 926 D.5. Aspects of constructive mathematics......Page 986 D.6. The logic of topoi......Page 1066 D.7. The type free lambda calculus......Page 1104 D.8. A mathematical incompleteness in Peano Arithmetic......Page 1146 Author Index......Page 1156 Subject Index......Page 1164 HANDBOOK OF MATHEMATICAL LOGIC by Jon Barwise [Hardcover] 1165 pages 072042285x The Handbook of Mathematical Logic is an attempt to share with the entire mathematical community some modern developments in logic. We have selected from the wealth of topics available some of those which deal with the basic concerns of the subject, or are particularly important for applications to other parts of mathematics, or both. Mathematical logic is traditionally divided into four model theory, set theory, recursion theory and proof theory. We have followed this division, for lack of a better one, in arranging this book. It made the placement of chapters where there is interaction of several parts of logic a difficult matter, so the division should be taken with a grain of salt. Each of the four parts begins with a short guide to the chapters that follow. The first chapter or two in each part are introductory in scope. More advanced chapters follow, as do chapters on applied or applicable parts of mathematical logic. Each chapter is definitely written for someone who is not a specialist in the field in question. On the other hand, each chapter has its own intended audience which varies from chapter to chapter. In particular, there are some chapters which are not written for the general mathematician, but rather are aimed at logicians in one field by logicians in another. We hope that many mathematicians will pick up this book out of idle curiosity and leaf through it to get a feeling for what is going on in another part of mathematics. It is hard to imagine a mathematician who could spend ten minutes doing this without wanting to pursue a few chapters, and the introductory sections of others, in some detail. It is an opportunity that hasnt existed before and is reason for the Handbook. -- Jon Barwise Edited By Jon Barwise, With The Cooperation Of H. J. Keisler ... [et Al]. Includes Bibliographies And Indexes.