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This book provides an authoritative and multifaceted introduction to eight major approaches to computation on uncountable mathematical domains. The perspectives explored within reveal different aspects of effective uncountable mathematics, making it an ideal resource for graduate and advanced undergraduate students and researchers in this exciting new area of study.

Aimed at graduate students and researchers in combinatorics and model theory, these two volumes provide a systematic exposition of methods used to classify homogeneous structures and the associated structural Ramsey theory of ordered graphs, graphs equipped with their natural metric structure, and similar structures.

Large Cardinals, Determinacy and Other Topics is the last in a series of four books presenting the seminal papers from the Caltech-UCLA 'Cabal Seminar' with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes.

This monograph is an exposition of the very general approach to the algebraization of sentential logics developed by the Barcelona logic group in the 1970s. It shows the results of this approach on a number of particular logics, and relates it to those developed by Blok, Czelakowski, Pigozzi and others.

Large cardinal hypotheses play a central role in modern set theory. This volume develops a method for constructing core models that have Woodin cardinals, a large cardinal hypothesis that is the focus of much current research. It is suitable for advanced graduate students and researchers in set theory.

This volume presents the conference proceedings of the 1995 Annual European Summer Meeting of the Association of Symbolic Logic. The papers here cover all topics of classical mathematical logic. There is also a tutorial on recent developments in theoretical computer science, and plenary lectures on the foundation and evolution of set theory.

This volume presents the proceedings of the 1996 European Summer Meeting of the Association for Symbolic Logic, held in San Sebastian, Spain. The main topics were model theory, proof theory, recursion and complexity theory, models of arithmetic, logic for artificial intelligence, formal semantics of natural language, and philosophy of contemporary logic.

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. In this volume, the third publication in the Lecture Notes in Logic series, Mitchell and Steel construct an inner model with a Woodin cardinal and develop its fine structure theory. This work builds upon the existing theory of a model of the form L[E], where E is a coherent sequence of extenders, and relies upon the fine structure theory of L[E] models with strong cardinals, and the theory of iteration trees and 'backgrounded' L[E] models with Woodin cardinals. This work is what results when fine structure meets iteration trees.

The proceedings of the Association for Symbolic Logic meeting held in Helsinki, Finland, in July 1990, containing eighteen papers written by leading researchers in logic. Between them they cover all fields of mathematical logic, including model theory, proof theory, recursion theory, and set theory.

The proceedings of the conference 'Logical Foundations of Mathematics, Computer Science, and Physics - Kurt Goedel's Legacy', held in Brno, Czech Republic, on the 90th anniversary of Goedel's birth. The papers in this volume cover the wide range of topics Goedel's work touched, and affirm its continuing importance.

This book is the first modern introduction to the logic of infinitary languages in forty years, and is aimed at graduate students and researchers in all areas of mathematical logic. Connections between infinitary model theory and other branches of mathematical logic, and applications to algebra and algebraic geometry are both comprehensively explored.

In this international cross-disciplinary collaboration, leading experts explore connections across their respective fields. The book offers an insight into the fundamental questions in quantum theory and logic, which will appeal to many researchers and graduate students including logicians, mathematicians, computer scientists, and physicists.

Ordinal Definability and Recursion Theory is the third in a series of four books presenting the seminal papers from the Caltech-UCLA 'Cabal Seminar' with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes.

This book, the first on the rapidly expanding topic of NIP theories, gives an accessible introduction to the subject for students and researchers in model theory and related areas such as combinatorics and algebraic geometry. It covers the basic notions while presenting a concise, elegant tour through the main results.

The Annual European Meeting of the Association for Symbolic Logic is among the most prestigious annual meetings in the field. Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world.

Alan Turing, whose centenary was celebrated worldwide in 2012, has come to be recognised as a genius of modern mathematics. This volume presents a diverse collection of essays, written by leading experts, on the many areas of logic and computer science that have their provenance in his work.

This volume surveys many exciting recent developments in the field of algorithmic randomness and its interactions with other areas of mathematics, presenting a unified view of the theory. It will be a valuable reference for specialists and an excellent entry point for graduate students and other newcomers to the field.

Papers examining aspects of Godel's work gathered from a symposium celebrating Goedel's centennial and papers from a 2004 ASL symposium.

This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow. The book also contains over 200 exercises, many with solutions, making the book a useful resource for graduate students as well as researchers.

Wadge Degrees and Projective Ordinals is the second of a series of four books presenting the seminal papers from the Caltech-UCLA 'Cabal Seminar' with extensive unpublished material, new papers on related topics and discussion of research developments since the publication of the original volumes.

The Annual European Meeting of the Association for Symbolic Logic is among the most prestigious annual meetings in the field. Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world.

This book presents a unifying framework for using priority arguments to prove theorems in computability.

Papers examining aspects of Godel's work gathered from a symposium celebrating Goedel's centennial and papers from a 2004 ASL symposium.

Games, Scales, and Suslin Cardinals is the first of a series of four books presenting the seminal papers from the Caltech-UCLA 'Cabal Seminar' with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes.

This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.

This book is an up-to-date introduction to simple theories and hyperimaginaries, with special attention to Lascar strong types and elimination of hyperimaginary problems. Assuming only knowledge of general model theory, the foundations of forking, stability and simplicity are presented in full detail. The treatment of the topics is as general as possible, working with stable formulas and types and assuming stability or simplicity of the theory only when necessary. The author offers an introduction to independence relations as well as a full account of canonical bases of types in stable and simple theories. In the last chapters the notions of internality and analyzability are discussed and used to provide a self-contained proof of elimination of hyperimaginaries in supersimple theories.

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. In this volume, the tenth publication in the Lecture Notes in Logic series, Per Lindstrom presents some of the main topics and results in general metamathematics. In addition to standard results of Godel et al. on incompleteness, (non-)finite axiomatizability, and interpretability, this book contains a thorough treatment of partial conservativity and degrees of interpretability. It comes complete with exercises, and will be useful as a textbook for graduate students with a background in logic, as well as a valuable resource for researchers.

The Annual European Meeting of the Association for Symbolic Logic, generally known as the Logic Colloquium, is the most prestigious annual meeting in the field. Many of the papers presented there are invited surveys of developments, and the rest of the papers are chosen to complement the invited talks. This 2007 volume includes surveys, tutorials, and selected research papers from the 2005 meeting. Highlights include three papers on different aspects of connections between model theory and algebra; a survey of major advances in combinatorial set theory; a tutorial on proof theory and modal logic; and a description of Bernay's philosophy of mathematics.

The Annual European Meeting of the Association for Symbolic Logic - or Logic Colloquium - is among the most prestigious annual meetings in the field. Logic Colloquium 2006, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world.

Highlights of this volume from the 2004 meeting include a tutorial survey of the recent highpoints of universal algebra, written by a leading expert; explorations of foundational questions; and a quartet of model theory papers giving an excellent reflection of current work in model theory, from the most abstract aspect 'abstract elementary classes' to issues around p-adic integration.