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Presents Results from a Very Active Area of ResearchExploring 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 mathem
Descriptive Set Theory by Yiannis N. Moschovakis Pdf
Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern ``effective'' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory.
Recursive Aspects of Descriptive Set Theory by Richard Mansfield,Galen Weitkamp Pdf
Explores the nature of infinity with a view toward classifying and explaining its mathematical applications. It presents not only the basics of the classical theory, but also an introduction to the many important recent results and methods.
Descriptive Set Theory and the Structure of Sets of Uniqueness by A. S. Kechris,Alain Louveau Pdf
To make this work accessible to logicians as well as set theorists and analysts, classical and modern theory of sets of uniqueness are covered as well as the relevant parts of descriptive set theory.
The Descriptive Set Theory of Polish Group Actions by Howard Becker,A. S. Kechris Pdf
In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.
Classical Descriptive Set Theory by Alexander Kechris Pdf
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.
Descriptive Set Theory and Definable Forcing by Jindřich Zapletal Pdf
Focuses on the relationship between definable forcing and descriptive set theory; the forcing serves as a tool for proving independence of inequalities between cardinal invariants of the continuum.
Generalized Descriptive Set Theory and Classification Theory by Sy D. Friedman,Tapani Hyttinen,Vadim Kulikov Pdf
Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper the authors study the generalization where countable is replaced by uncountable. They explore properties of generalized Baire and Cantor spaces equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical countable case. They also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. The authors' results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.
Generalized Descriptive Set Theory and Classification Theory by Sy-David Friedman,Tapani Hyttinen, Vadim Kulikov Pdf
Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper the authors study the generalization where countable is replaced by uncountable. They explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. They also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. The authors' results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.
Descriptive Set Theory and Dynamical Systems by M. Foreman Pdf
This volume, first published in 2000, contains a collection of survey papers providing an introduction for graduate students and researchers in these fields.
