Perspectives in Mathematical Logic

This book series is now being published by the Association for Symbolic Logic on its own; the previous collaboration with Springer-Verlag came to an end on April 30, 2001. Thanks to the generosity of Springer-Verlag, ASL will distribute the available stock of certain books in the series to the logic community at a low price (as has been done with the existing stock of books in the Lecture Notes in Logic). Some books in the series will continue to be made available by Springer-Verlag and others will be reprinted by ASL. At the moment (October 2001) the situation is in flux and plans for the future are being made. Inquiries may be made via the ASL business office :

Association for Symbolic Logic
Box 742, Vassar College
124 Raymond Avenue
Poughkeepsie, New York 12604
USA
Email: asl@vassar.edu
telephone: 845-437-7080 Fax: 845-437-7830

Books are listed in reverse order by date of publication.

This book is an original contribution to the foundations of mathematics, with emphasis on the role of set existence axioms. Part A demonstrates that many familiar theorems of algebra, analysis, functional analysis, and combinatorics are logically equivalent to the axioms needed to prove them. This phenomenon is known as Reverse Mathematics. Subsystems of second order arithmetic based on such axioms correspond to several well known foundational programs: finitistic reductionism (Hilbert), constructivism (Bishop), predicativism (Weyl), and predicative reductionism (Feferman/Friedman). Part B is a thorough study of models of these and other systems. The book includes an extensive bibliography and a detailed index.

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Finite model theory has its origin in classical model theory, but owes its systematic development to research from complexity theory. This book presents the main results of descriptive complexity theory, that is, the connections between axiomatizability of classes of finite structures and their complexity with respect to time and space bounds. The logics that are important in the context include fixed-point logics, transitive closure logics, and also certain infinitary languages; their model theory is studied in full detail. Other topics include DATALOG languages, quantifiers and oracles, 0-1 laws, and optimization and approximation problems. The book is written is such a way that the respective parts on model theory and descriptive complexity theory may be read independently.

For the second edition, the original text was thoroughly revised and extended; in particular, a new chapter was added which is devoted to a central open problem of finite model theory, namely the question whether there is a natural logic capturing PTIME also on unordered structures. The bibliography and the bibliographic references at the end of each chapter were also enlarged considerably.

This book deals with set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. Consequently, the theory of iterated forcing is developed. The author aims at giving a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations. In addition to particular consistency results, the author shows methods which can be used for such independence results. Many of the results are presented in an "axiomatic" framework (a la Martin's axiom) for this reason.

Second edition 1998/1064 pages/Hardcover/ISBN 3-540-51700-6/$189.00

This book covers the most important results in the study of a first order theory of the natural numbers, called Peano arithmetic, and its subsystems. The first part develops naturally important parts of mathematics and logic in suitable fragments. The second, is devoted to incompleteness and the third, studies systems that have the induction schema restricted to bounded formulas. A highlight of this section is the relation of provability to computational complexity. Intended for those who want to learn more about formal systems for arithmetic and the current research in the field.

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This is the most complete and comprehensive treatment available in book form of the classical decision problem of mathematical logic and its role in modern computer science. A revealing analysis of the natural order of decidable and undecidable cases is given, as well as the complete classification of the solvable and unsolvable standard cases of the classical decision problem, the complexity analysis of the solvable cases, the extremely comprehensive treatment of the reduction method, and the model-theoretic analysis of solvable cases. Many cases are treated here for the first time, and a great number of simple proofs and exercises are included.

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This book is the standard reference on set theory covering all aspects of its two main general areas: classical set theory including large cardinals, infinitary combinatorics, descriptive set theory, and independence proofs beginning with the work of G”del around 1938 and continuing with the work of Cohen in 1963 which led to many important applications to mathematics. The author's primary emphasis is on forcing and large cardinals (on which the author has collected an enormous amount of material that has previously only been available through scattered journal articles or private communication) but there is also a substantial discussion of descriptive set theory and infinitary combinatorics as well. This very well-organized presentation has become both a widely used textbook, and a standard reference.

Stability theory began in the early 1960s with the work of Michael Morley and matured in the 70s through Shelah's research in model-theoretic classification theory. Today stability theory both influences and is influenced by number theory, algebraic group theory, Riemann surfaces, and representation theory of modules. there is little model theory today that does not involve the methods of stability theory. The aim of this book is to provide the student with a quick route from basic model theory to research in stability theory, to prepare a student for research in any of today's branches of stability theory and to give an introduction to classification theory with an exposition of Morley's Categoricity Theorem.

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"[A]n excellent and thorough treatment of that part of the subject which lies closest to its origins in the work of Gödel, Church, Kleene, and Turing in the 1930s. The focus is almost exclusively on functions and sets of natural numbers and more particularly on those sets which are either recursively enumerable or whose nature is somehow determined by recursive enumerability." Mathematical Reviews