Download or read book Introduction to Cardinal Arithmetic written by Michael Holz and published by Springer Science & Business Media. This book was released on 2009-11-23 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to modern cardinal arithmetic, developed in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice. It splits into three parts. Part one, which is contained in Chapter 1, describes the classical cardinal arithmetic due to Bernstein, Cantor, Hausdorff, Konig, and Tarski. The results were found in the years between 1870 and 1930. Part two, which is Chapter 2, characterizes the development of cardinal arith metic in the seventies, which was led by Galvin, Hajnal, and Silver. The third part, contained in Chapters 3 to 9, presents the fundamental investigations in pcf-theory which has been developed by S. Shelah to answer the questions left open in the seventies. All theorems presented in Chapter 3 and Chapters 5 to 9 are due to Shelah, unless otherwise stated. We are greatly indebted to all those set theorists whose work we have tried to expound. Concerning the literature we owe very much to S. Shelah's book [Sh5] and to the article by M. R. Burke and M. Magidor [BM] which also initiated our students' interest for Shelah's pcf-theory.
Download or read book Introduction to Cardinal Arithmetic written by Michael Holz and published by Springer Science & Business Media. This book was released on 1999-09 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to modern cardinal arithmetic is presented in this volume, in addition to a survey of results. A discussion of classical theory is included, paired with investigations in pcf theory, which answers questions left open since the 1970’s.
Download or read book Introduction to Cardinal Arithmetic written by Michael Holz and published by Birkhäuser. This book was released on 2010-04-06 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to modern cardinal arithmetic, developed in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice. It splits into three parts. Part one, which is contained in Chapter 1, describes the classical cardinal arithmetic due to Bernstein, Cantor, Hausdorff, Konig, and Tarski. The results were found in the years between 1870 and 1930. Part two, which is Chapter 2, characterizes the development of cardinal arith metic in the seventies, which was led by Galvin, Hajnal, and Silver. The third part, contained in Chapters 3 to 9, presents the fundamental investigations in pcf-theory which has been developed by S. Shelah to answer the questions left open in the seventies. All theorems presented in Chapter 3 and Chapters 5 to 9 are due to Shelah, unless otherwise stated. We are greatly indebted to all those set theorists whose work we have tried to expound. Concerning the literature we owe very much to S. Shelah's book [Sh5] and to the article by M. R. Burke and M. Magidor [BM] which also initiated our students' interest for Shelah's pcf-theory.
Download or read book Cardinal Arithmetic written by Saharon Shelah and published by Oxford University Press on Demand. This book was released on 1994 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is the continuum hypothesis still open? If we interpret it as finding the laws of cardinal arithmetic (really exponentiation since addition and multiplication were classically solved), it was thought to be essentially solved by the independence results of Godel and Cohen (and Easton) with some isolated positive results (likeGalvin-Hajnal). It was expected that only more independence results remained to be proved. The author has come to change his view. This enables us to get new results for the conventional cardinal arithmetic, thus supporting the interest in our view. We also find other applications, extend older methods of using normal fiters and prove the existence of Jonsson algebra.
Download or read book Set Theory written by Lev D. Beklemishev and published by Elsevier. This book was released on 2000-04-01 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: Set Theory
Download or read book A Course on Borel Sets written by S.M. Srivastava and published by Springer Science & Business Media. This book was released on 1998-04-13 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thorough introduction to Borel sets and measurable selections, acting as a stepping stone to descriptive set theory by presenting such important techniques as universal sets, prewellordering, scales, etc. It contains significant applications to other branches of mathematics and serves as a self-contained reference accessible by mathematicians in many different disciplines. Written in an easily understandable style, and using only naive set theory, general topology, analysis, and algebra, it is thus well suited for graduates exploring areas of mathematics for their research and for those requiring Borel sets and measurable selections in their work.
Download or read book Principia Mathematica written by Alfred North Whitehead and published by . This book was released on 1910 with total page 688 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Handbook of Set Theory written by Matthew Foreman and published by Springer Science & Business Media. This book was released on 2009-12-10 with total page 2200 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Introduction to Cardinal Arithmetic written by Michael Holz and published by Birkhauser. This book was released on 1999-01-01 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Cardinal Algebras written by Alfred Tarski and published by . This book was released on 1949 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Set Theory and its Philosophy written by Michael Potter and published by Clarendon Press. This book was released on 2004-01-15 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.
Download or read book Numbers Sets and Axioms written by A. G. Hamilton and published by Cambridge University Press. This book was released on 1982 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: Following the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required.
Download or read book Cardinal Arithmetic written by S. Shelah and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Setting a new direction in research in the subject, this book presents a new view of cardinal arithmetic, one of the central issues in set theory. Focusing on cofinalities rather than cardinalities, new results are obtained and published here for the first time.
Download or read book The Theory of Sets and Transfinite Arithmetic written by Alexander Abian and published by . This book was released on 1965 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Encyclopaedia Britannica Lor to Mun written by and published by . This book was released on 1911 with total page 1056 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Encyclop dia Britannica written by Hugh Chisholm and published by . This book was released on 1911 with total page 1058 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Encyclopaedia Britannica Lord Chamberlain Mecklenberg written by and published by . This book was released on 1911 with total page 1070 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The last great work of the age of reason, the final instance when all human knowledge could be presented with a single point of view ... Unabashed optimism, and unabashed racism, pervades many entries in the 11th, and provide its defining characteristics ... Despite its occasional ugliness, the reputation of the 11th persists today because of the staggering depth of knowledge contained with its volumes. It is especially strong in its biographical entries. These delve deeply into the history of men and women prominent in their eras who have since been largely forgotten - except by the historians, scholars"-- The Guardian, https://www.theguardian.com/books/booksblog/2012/apr/10/encyclopedia-britannica-11th-edition.
