Classical Descriptive Set Theory

Author: Alexander Kechris
Publisher: Springer Science & Business Media
ISBN: 9781461241904
Release Date: 2012-12-06
Genre: Mathematics

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text presents a largely balanced approach to the subject, which combines many elements of the different traditions. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.

Invariant Descriptive Set Theory

Author: Su Gao
Publisher: CRC Press
ISBN: 158488794X
Release Date: 2008-09-03
Genre: Mathematics

Presents Results from a Very Active Area of Research Exploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas of mathematics, such as algebra, topology, and logic, which have diverse applications to other fields. After reviewing classical and effective descriptive set theory, the text studies Polish groups and their actions. It then covers Borel reducibility results on Borel, orbit, and general definable equivalence relations. The author also provides proofs for numerous fundamental results, such as the Glimm–Effros dichotomy, the Burgess trichotomy theorem, and the Hjorth turbulence theorem. The next part describes connections with the countable model theory of infinitary logic, along with Scott analysis and the isomorphism relation on natural classes of countable models, such as graphs, trees, and groups. The book concludes with applications to classification problems and many benchmark equivalence relations. By illustrating the relevance of invariant descriptive set theory to other fields of mathematics, this self-contained book encourages readers to further explore this very active area of research.

Banach Spaces and Descriptive Set Theory Selected Topics

Author: Pandelis Dodos
Publisher: Springer Science & Business Media
ISBN: 9783642121524
Release Date: 2010-05-10
Genre: Mathematics

This volume deals with problems in the structure theory of separable infinite-dimensional Banach spaces, with a central focus on universality problems. This topic goes back to the beginnings of the field and appears in Banach's classical monograph. The novelty of the approach lies in the fact that the answers to a number of basic questions are based on techniques from Descriptive Set Theory. Although the book is oriented on proofs of several structural theorems, in the main text readers will also find a detailed exposition of numerous “intermediate” results which are interesting in their own right and have proven to be useful in other areas of Functional Analysis. Moreover, several well-known results in the geometry of Banach spaces are presented from a modern perspective.

Dynamics of Linear Operators

Author: Frédéric Bayart
Publisher: Cambridge University Press
ISBN: 9780521514965
Release Date: 2009-06-04
Genre: Mathematics

The first book to assemble the wide body of theory which has rapidly developed on the dynamics of linear operators. Written for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.

Zahlen

Author: Heinz-Dieter Ebbinghaus
Publisher: Springer-Verlag
ISBN: 9783642971228
Release Date: 2013-03-08
Genre: Mathematics

Aus den Besprechungen: "Ein Mathematikbuch der Superlativen, für Mathematiker (jeder Schattierung) und Nichtmathematiker (denen völlig unbekannte Dimensionen der Mathematik eröffnet werden - künstlerische, magische, historische, philosophische, wissenschaftstheoretische, "unlogische", phantasieerfüllte usw.). Der Aufbau ist meisterhaft, die Lektüre höchst anregend und leicht lesbar." Monatshefte für Mathematik #1 "Ein gelungenes Werk, das dem Vorurteil entgegenwirkt, Mathematik bestehe nur aus isolierten Theorien." Die NEUE HOCHSCHULE #1 "Das Lesen ist ein Genuß, den man sich nicht entgehen lassen sollte." Jahresbericht der Deutschen Mathematiker-Vereinigung #1

Fundamentals of Mathematical Logic

Author: Peter G. Hinman
Publisher: A K Peters/CRC Press
ISBN: 1568812620
Release Date: 2005-09-09
Genre: Mathematics

This introductory graduate text covers modern mathematical logic from propositional, first-order and infinitary logic and Gödel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching experience, the book develops students' intuition by presenting complex ideas in the simplest context for which they make sense. The book is appropriate for use as a classroom text, for self-study, and as a reference on the state of modern logic.

Ma und Kategorie

Author: J.C. Oxtoby
Publisher: Springer-Verlag
ISBN: 9783642960741
Release Date: 2013-03-08
Genre: Mathematics

Dieses Buch behandelt hauptsächlich zwei Themenkreise: Der Bairesche Kategorie-Satz als Hilfsmittel für Existenzbeweise sowie Die "Dualität" zwischen Maß und Kategorie. Die Kategorie-Methode wird durch viele typische Anwendungen erläutert; die Analogie, die zwischen Maß und Kategorie besteht, wird nach den verschiedensten Richtungen hin genauer untersucht. Hierzu findet der Leser eine kurze Einführung in die Grundlagen der metrischen Topologie; außerdem werden grundlegende Eigenschaften des Lebesgue schen Maßes hergeleitet. Es zeigt sich, daß die Lebesguesche Integrationstheorie für unsere Zwecke nicht erforderlich ist, sondern daß das Riemannsche Integral ausreicht. Weiter werden einige Begriffe aus der allgemeinen Maßtheorie und Topologie eingeführt; dies geschieht jedoch nicht nur der größeren Allgemeinheit wegen. Es erübrigt sich fast zu erwähnen, daß sich die Bezeichnung "Kategorie" stets auf "Bairesche Kategorie" be zieht; sie hat nichts zu tun mit dem in der homologischen Algebra verwendeten Begriff der Kategorie. Beim Leser werden lediglich grundlegende Kenntnisse aus der Analysis und eine gewisse Vertrautheit mit der Mengenlehre vorausgesetzt. Für die hier untersuchten Probleme bietet sich in natürlicher Weise die mengentheoretische Formulierung an. Das vorlie gende Buch ist als Einführung in dieses Gebiet der Analysis gedacht. Man könnte es als Ergänzung zur üblichen Grundvorlesung über reelle Analysis, als Grundlage für ein Se minar oder auch zum selbständigen Studium verwenden. Bei diesem Buch handelt es sich vorwiegend um eine zusammenfassende Darstellung; jedoch finden sich in ihm auch einige Verfeinerungen bekannter Resultate, namentlich Satz 15.6 und Aussage 20.4. Das Literaturverzeichnis erhebt keinen Anspruch auf Vollständigkeit. Häufig werden Werke zitiert, die weitere Literaturangaben enthalten.

Logic Computation Hierarchies

Author: Vasco Brattka
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 9781614519409
Release Date: 2014-09-04
Genre: Philosophy

Published in honor of Victor L. Selivanov, the 17 articles collected in this volume inform on the latest developments in computability theory and its applications in computable analysis; descriptive set theory and topology; and the theory of omega-languages; as well as non-classical logics, such as temporal logic and paraconsistent logic. This volume will be of interest to mathematicians and logicians, as well as theoretical computer scientists.

Set theory

Author: Alessandro Andretta
Publisher:
ISBN: UOM:39015069032350
Release Date: 2005
Genre: Mathematics


Logic Colloquium 01

Author: Matthias Baaz
Publisher: A K Peters Ltd
ISBN: CORNELL:31924100528961
Release Date: 2005-06-15
Genre: Mathematics

A compilation of papers presented at the 2001 European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium '01 includes surveys and research articles from some of the world's preeminent logicians. Two long articles are based on tutorials given at the meeting and present accessible expositions of research in two active areas of logic, geometric model theory and descriptive set theory of group actions. The remaining articles cover seperate research topics in many areas of mathematical logic, including applications in Computer Science, Proof Theory, Set Theory, Model Theory, Computability Theory, and aspects of Philosophy. This collection will be of interest to philosophical logicians, historians of logic, computer scientists, formal linguists, and mathematicians in the areas of algebra, abstract analysis and topology.