Universality In Set Theories
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Universality In Set Theories
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Author : Manuel Bremer
language : en
Publisher: Walter de Gruyter
Release Date : 2013-05-02
Universality In Set Theories written by Manuel Bremer and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-05-02 with Philosophy categories.
The book discusses the fate of universality and a universal set in several set theories. The book aims at a philosophical study of ontological and conceptual questions around set theory. Set theories are ontologies. They posit sets and claim that these exhibit the essential properties laid down in the set theoretical axioms. Collecting these postulated entities quantified over poses the problem of universality. Is the collection of the set theoretical entities itself a set theoretical entity? What does it mean if it is, and what does it mean if it is not? To answer these questions involves developing a theory of the universal set. We have to ask: Are there different aspects to universality in set theory, which stand in conflict to each other? May inconsistency be the price to pay to circumvent ineffability? And most importantly: How far can axiomatic ontology take us out of the problems around universality?
Set Theory With A Universal Set
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Author : T. E. Forster
language : en
Publisher: Oxford University Press on Demand
Release Date : 1995
Set Theory With A Universal Set written by T. E. Forster and has been published by Oxford University Press on Demand this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.
Set theory is concerned with the foundation of mathematics. In the original formulations of set theory, there were paradoxes contained in the idea of the "set of all sets". Current standard theory (Zermelo-Fraenkel) avoids these paradoxes by restricting the way sets may be formed by othersets, specifically to disallow the possibility of forming the set of all sets. In the 1930s, Quine proposed a different form of set theory in which the set of all sets - the universal set - is allowed, but other restrictions are placed on these axioms. Since then, the steady interest expressed inthese non-standard set theories has been boosted by their relevance to computer science.The second edition still concentrates largely on Quine's New Foundations, reflecting the author's belief that this provides the richest and most mysterious of the various systems dealing with set theories with a universal set. Also included is an expanded and completely revised account of the settheories of Church-Oswald and Mitchell, with descriptions of permutation models and extensions that preserve power sets. Dr Foster here presents the reader with a useful and readable introduction for those interested in this topic, and a reference work for those already involved in this area.
A Book Of Set Theory
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Author : Charles C Pinter
language : en
Publisher: Courier Corporation
Release Date : 2014-07-23
A Book Of Set Theory written by Charles C Pinter and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-23 with Mathematics categories.
"This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author"--
Fundamentals Of Set And Number Theory
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Author : Valeriy K. Zakharov
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-02-05
Fundamentals Of Set And Number Theory written by Valeriy K. Zakharov and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-02-05 with Mathematics categories.
This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics.The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Fundamentals of the theory of classes, sets, and numbers Characterization of all natural models of Neumann – Bernays – Godel and Zermelo – Fraenkel set theories Local theory of sets as a foundation for category theory and its connection with the Zermelo – Fraenkel set theory Compactness theorem for generalized second-order language
Fuzzy Set Theory
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Author : Michael Smithson
language : en
Publisher: SAGE
Release Date : 2006-02-17
Fuzzy Set Theory written by Michael Smithson and has been published by SAGE this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-17 with Mathematics categories.
Dealing with sets or categories whose boundaries are blurry or 'fuzzy', this book is an accessible introduction to fuzzy set theory, focusing on its applicability to the social sciences.
Set Theory
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Author : Thomas Jech
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-05-23
Set Theory written by Thomas Jech 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 2007-05-23 with Mathematics categories.
Set Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as a desktop reference.
Classical Descriptive Set Theory
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Author : Alexander Kechris
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Classical Descriptive Set Theory written by Alexander Kechris 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 2012-12-06 with Mathematics categories.
Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.
Encyclopaedia Of Mathematics
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Author : Michiel Hazewinkel
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-01
Encyclopaedia Of Mathematics written by Michiel Hazewinkel 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 2013-12-01 with Mathematics categories.
This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.
Finsler Set Theory Platonism And Circularity
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Author : David Booth
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Finsler Set Theory Platonism And Circularity written by David Booth and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Finsler's papers on set theory are presented, here for the first time in English translation, in three parts, and each is preceded by an introduction to the field written by the editors. In the philosophical part of his work Finsler develops his approach to the paradoxes, his attitude toward formalized theories and his defense of Platonism in mathematics. He insisted on the existence of a conceptual realm within mathematics that transcends formal systems. From the foundational point of view, Finsler's set theory contains a strengthened criterion for set identity and a coinductive specification of the universe of sets. The notion of the class of circle free sets introduced by Finsler is potentially a very fertile one although not very widespread today. Combinatorially, Finsler considers sets as generalized numbers to which one may apply arithmetical techniques. The introduction to this third section of the book extends Finsler's theory to non-well-founded sets. The present volume makes Finsler's papers on set theory accessible at long last to a wider group of mathematicians, philosophers and historians of science. A technical background is not necessary to appreciate the satisfying interplay of philosophical and mathematical ideas that characterizes this work.
An Introduction To Proofs With Set Theory
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Author : Daniel Ashlock
language : en
Publisher: Springer Nature
Release Date : 2022-06-01
An Introduction To Proofs With Set Theory written by Daniel Ashlock and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-06-01 with Mathematics categories.
This text is intended as an introduction to mathematical proofs for students. It is distilled from the lecture notes for a course focused on set theory subject matter as a means of teaching proofs. Chapter 1 contains an introduction and provides a brief summary of some background material students may be unfamiliar with. Chapters 2 and 3 introduce the basics of logic for students not yet familiar with these topics. Included is material on Boolean logic, propositions and predicates, logical operations, truth tables, tautologies and contradictions, rules of inference and logical arguments. Chapter 4 introduces mathematical proofs, including proof conventions, direct proofs, proof-by-contradiction, and proof-by-contraposition. Chapter 5 introduces the basics of naive set theory, including Venn diagrams and operations on sets. Chapter 6 introduces mathematical induction and recurrence relations. Chapter 7 introduces set-theoretic functions and covers injective, surjective, and bijective functions, as well as permutations. Chapter 8 covers the fundamental properties of the integers including primes, unique factorization, and Euclid's algorithm. Chapter 9 is an introduction to combinatorics; topics included are combinatorial proofs, binomial and multinomial coefficients, the Inclusion-Exclusion principle, and counting the number of surjective functions between finite sets. Chapter 10 introduces relations and covers equivalence relations and partial orders. Chapter 11 covers number bases, number systems, and operations. Chapter 12 covers cardinality, including basic results on countable and uncountable infinities, and introduces cardinal numbers. Chapter 13 expands on partial orders and introduces ordinal numbers. Chapter 14 examines the paradoxes of naive set theory and introduces and discusses axiomatic set theory. This chapter also includes Cantor's Paradox, Russel's Paradox, a discussion of axiomatic theories, an exposition on Zermelo‒Fraenkel Set Theory with the Axiom of Choice, and a brief explanation of Gödel's Incompleteness Theorems.