Surveys In Set Theory
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Surveys In Set Theory
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Author : A. R. D. Mathias
language : en
Publisher: Cambridge University Press
Release Date : 1983-10-13
Surveys In Set Theory written by A. R. D. Mathias and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1983-10-13 with Mathematics categories.
This book comprises five expository articles and two research papers on topics of current interest in set theory and the foundations of mathematics. Articles by Baumgartner and Devlin introduce the reader to proper forcing. This is a development by Saharon Shelah of Cohen's method which has led to solutions of problems that resisted attack by forcing methods as originally developed in the 1960s. The article by Guaspari is an introduction to descriptive set theory, a subject that has developed dramatically in the last few years. Articles by Kanamori and Stanley discuss one of the most difficult concepts in contemporary set theory, that of the morass, first created by Ronald Jensen in 1971 to solve the gap-two conjecture in model theory, assuming Gödel's axiom of constructibility. The papers by Prikry and Shelah complete the volume by giving the reader the flavour of contemporary research in set theory. This book will be of interest to graduate students and research workers in set theory and mathematical logic.
Descriptive Set Theory
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Author : Yiannis N. Moschovakis
language : en
Publisher: American Mathematical Society
Release Date : 2025-01-31
Descriptive Set Theory written by Yiannis N. Moschovakis and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-31 with Mathematics categories.
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.
Set Theory
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Author : Simon Thomas
language : en
Publisher: American Mathematical Soc.
Release Date : 2002
Set Theory written by Simon Thomas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.
This volume presents the proceedings from the Mid-Atlantic Mathematical Logic Seminar (MAMLS) conference held in honor of Andras Hajnal at the DIMACS Center, Rutgers University (New Brunswick, NJ). Articles include both surveys and high-level research papers written by internationally recognized experts in the field of set theory. Many of the current active areas of set theory are represented in this volume. It includes research papers on combinatorial set theory, set theoretictopology, descriptive set theory, and set theoretic algebra. There are valuable surveys on combinatorial set theory, fragments of the proper forcing axiom, and the reflection properties of stationary sets. The book also includes an exposition of the ergodic theory of lattices in higher rank semisimpleLie groups-essential reading for anyone who wishes to understand much of the recent work on countable Borel equivalence relations.
The Descriptive Set Theory Of Polish Group Actions
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Author : Howard Becker
language : en
Publisher: Cambridge University Press
Release Date : 1996-12-05
The Descriptive Set Theory Of Polish Group Actions written by Howard Becker and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-12-05 with Mathematics categories.
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.
Labyrinth Of Thought
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Author : Jose Ferreiros
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-11-01
Labyrinth Of Thought written by Jose Ferreiros and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-11-01 with Mathematics categories.
"José Ferreirós has written a magisterial account of the history of set theory which is panoramic, balanced, and engaging. Not only does this book synthesize much previous work and provide fresh insights and points of view, but it also features a major innovation, a full-fledged treatment of the emergence of the set-theoretic approach in mathematics from the early nineteenth century. This takes up Part One of the book. Part Two analyzes the crucial developments in the last quarter of the nineteenth century, above all the work of Cantor, but also Dedekind and the interaction between the two. Lastly, Part Three details the development of set theory up to 1950, taking account of foundational questions and the emergence of the modern axiomatization." (Bulletin of Symbolic Logic)
Geometric Set Theory
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Author : Paul B. Larson
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-07-16
Geometric Set Theory written by Paul B. Larson and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-07-16 with Education categories.
This book introduces a new research direction in set theory: the study of models of set theory with respect to their extensional overlap or disagreement. In Part I, the method is applied to isolate new distinctions between Borel equivalence relations. Part II contains applications to independence results in Zermelo–Fraenkel set theory without Axiom of Choice. The method makes it possible to classify in great detail various paradoxical objects obtained using the Axiom of Choice; the classifying criterion is a ZF-provable implication between the existence of such objects. The book considers a broad spectrum of objects from analysis, algebra, and combinatorics: ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and many more. The topic is nearly inexhaustible in its variety, and many directions invite further investigation.
Quine New Foundations And The Philosophy Of Set Theory
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Author : Sean Morris
language : en
Publisher: Cambridge University Press
Release Date : 2018-12-13
Quine New Foundations And The Philosophy Of Set Theory written by Sean Morris and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-12-13 with History categories.
Provides an accessible mathematical and philosophical account of Quine's set theory, New Foundations.
Surveys In Combinatorics 2021
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Author : Konrad K. Dabrowski
language : en
Publisher: Cambridge University Press
Release Date : 2021-06-24
Surveys In Combinatorics 2021 written by Konrad K. Dabrowski and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-24 with Mathematics categories.
These nine articles provide up-to-date surveys of topics of contemporary interest in combinatorics.
Applications Of Point Set Theory In Real Analysis
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Author : A.B. Kharazishvili
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Applications Of Point Set Theory In Real Analysis written by A.B. Kharazishvili and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, W«;glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi
Handbook Of Set Theory
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Author : Matthew Foreman
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-12-10
Handbook Of Set Theory written by Matthew Foreman and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-12-10 with Mathematics categories.
Numbers imitate space, which is of such a di?erent nature —Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.