Uncountably Categorical Theories
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| Author | : Boris Zilber |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 132 |
| Release | : |
| Genre | : Mathematics |
| ISBN | : 9780821897454 |
The 1970s saw the appearance and development in categoricity theory of a tendency to focus on the study and description of uncountably categorical theories in various special classes defined by natural algebraic or syntactic conditions. There have thus been studies of uncountably categorical theories of groups and rings, theories of a one-place function, universal theories of semigroups, quasivarieties categorical in infinite powers, and Horn theories. In Uncountably Categorical Theories , this research area is referred to as the special classification theory of categoricity. Zilber's goal is to develop a structural theory of categoricity, using methods and results of the special classification theory, and to construct on this basis a foundation for a general classification theory of categoricity, that is, a theory aimed at describing large classes of uncountably categorical structures not restricted by any syntactic or algebraic conditions.
| Author | : Peter Cholak |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 338 |
| Release | : 2000 |
| Genre | : Mathematics |
| ISBN | : 0821819224 |
This collection of articles presents a snapshot of the status of computability theory at the end of the millennium and a list of fruitful directions for future research. The papers represent the works of experts in the field who were invited speakers at the AMS-IMS-SIAM 1999 Summer Conference on Computability Theory and Applications, which focused on open problems in computability theory and on some related areas in which the ideas, methods, and/or results of computability theory play a role. Some presentations are narrowly focused; others cover a wider area. Topics included from "pure" computability theory are the computably enumerable degrees (M. Lerman), the computably enumerable sets (P. Cholak, R. Soare), definability issues in the c.e. and Turing degrees (A. Nies, R. Shore) and other degree structures (M. Arslanov, S. Badaev and S. Goncharov, P. Odifreddi, A. Sorbi). The topics involving relations between computability and other areas of logic and mathematics are reverse mathematics and proof theory (D. Cenzer and C. Jockusch, C. Chong and Y. Yang, H. Friedman and S. Simpson), set theory (R. Dougherty and A. Kechris, M. Groszek, T. Slaman) and computable mathematics and model theory (K. Ambos-Spies and A. Kucera, R. Downey and J. Remmel, S. Goncharov and B. Khoussainov, J. Knight, M. Peretyat'kin, A. Shlapentokh).
| Author | : Steven Buechler |
| Publisher | : Cambridge University Press |
| Total Pages | : 368 |
| Release | : 2017-03-02 |
| Genre | : Mathematics |
| ISBN | : 1107168392 |
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Stability theory was introduced and matured in the 1960s and 1970s. Today stability theory 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. In this volume, the fourth publication in the Perspectives in Logic series, Steven Buechler bridges the gap between a first-year graduate logic course and research papers in stability theory. The book prepares the student for research in any of today's branches of stability theory, and gives an introduction to classification theory with an exposition of Morley's Categoricity Theorem.
| Author | : Dov Gabbay |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 377 |
| Release | : 2007-07-28 |
| Genre | : Mathematics |
| ISBN | : 0387692452 |
This book presents contributions from world-renowned logicians, discussing important topics of logic from the point of view of their further development in light of requirements arising from successful application in Computer Science and AI language. Coverage includes: the logic of provability, computability theory applied to biology, psychology, physics, chemistry, economics, and other basic sciences; computability theory and computable models; logic and space-time geometry; hybrid systems; logic and region-based theory of space.
| Author | : O.V. Belegradek |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 362 |
| Release | : 1999 |
| Genre | : Mathematics |
| ISBN | : 9780821896037 |
This volume is a collection of papers on model theory and its applications. The longest paper, "Model Theory of Unitriangular Groups" by O. V. Belegradek, forms a subtle general theory behind Mal'tsev's famous correspondence between rings and groups. This is the first published paper on the topic. Given the present model-theoretic interest in algebraic groups, Belegradek's work is of particular interest to logicians and algebraists. The rest of the collection consists of papers on various questions of model theory, mainly on stability theory. Contributors are leading Russian researchers in the field.
| Author | : David Marker |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 342 |
| Release | : 2006-04-06 |
| Genre | : Mathematics |
| ISBN | : 0387227342 |
Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures
| Author | : Noam Greenberg |
| Publisher | : Cambridge University Press |
| Total Pages | : 205 |
| Release | : 2013-10-31 |
| Genre | : Mathematics |
| ISBN | : 1107014514 |
A comprehensive introduction to eight major approaches to computation on uncountable mathematical domains.
| Author | : Wilfrid Hodges |
| Publisher | : Cambridge University Press |
| Total Pages | : 810 |
| Release | : 1993-03-11 |
| Genre | : Mathematics |
| ISBN | : 9780521304429 |
Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide range of other areas such as set theory, geometry, algebra and computer science. This book provides an integrated introduction to model theory for graduate students.
| Author | : Deirdre Haskell |
| Publisher | : Cambridge University Press |
| Total Pages | : 244 |
| Release | : 2000-07-03 |
| Genre | : Mathematics |
| ISBN | : 9780521780681 |
Model theory has made substantial contributions to semialgebraic, subanalytic, p-adic, rigid and diophantine geometry. These applications range from a proof of the rationality of certain Poincare series associated to varieties over p-adic fields, to a proof of the Mordell-Lang conjecture for function fields in positive characteristic. In some cases (such as the latter) it is the most abstract aspects of model theory which are relevant. This book, originally published in 2000, arising from a series of introductory lectures for graduate students, provides the necessary background to understanding both the model theory and the mathematics behind these applications. The book is unique in that the whole spectrum of contemporary model theory (stability, simplicity, o-minimality and variations) is covered and diverse areas of geometry (algebraic, diophantine, real analytic, p-adic, and rigid) are introduced and discussed, all by leading experts in their fields.
| Author | : Philipp Rothmaler |
| Publisher | : CRC Press |
| Total Pages | : 324 |
| Release | : 2018-12-07 |
| Genre | : Mathematics |
| ISBN | : 0429668503 |
Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.
