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Author : Sean Morris
Publisher : Cambridge University Press
Page : 221 pages
File Size : 53,9 Mb
Release : 2018-12-13
Category : History
ISBN : 9781107152502
Provides an accessible mathematical and philosophical account of Quine's set theory, New Foundations.
Author : Luca Incurvati
Publisher : Cambridge University Press
Page : 255 pages
File Size : 47,9 Mb
Release : 2020-01-23
Category : History
ISBN : 9781108497824
Presents a detailed and critical examination of the available conceptions of set and proposes a novel version.
Author : Willard Van Orman Quine
Publisher : Harvard University Press
Page : 384 pages
File Size : 40,5 Mb
Release : 1969
Category : Mathematics
ISBN : 0674802071
This is an extensively revised edition of W. V. Quine’s introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted.
Author : Alfred North Whitehead,Bertrand Russell
Publisher : Cambridge University Press
Page : 524 pages
File Size : 44,8 Mb
Release : 1927
Category : Mathematics
ISBN : 052106791X
The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century.
Author : Michael Potter
Publisher : Clarendon Press
Page : 362 pages
File Size : 43,5 Mb
Release : 2004-01-15
Category : Philosophy
ISBN : 9780191556432
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.
Author : L. Decock
Publisher : Springer Science & Business Media
Page : 282 pages
File Size : 40,7 Mb
Release : 2013-03-09
Category : Philosophy
ISBN : 9789401735759
Willard VanOrman Quine has probably been the most influential th American philosopher of the 20 century. His work spans over seven decades, and covers many domains in philosophy. He has made major contributions to the fields of logic and set theory, philosophy of logic and mathematics, philosophy of language, philosophy of science, epistemology and metaphysics. Quine's first work in philosophy was in the field of logic. His major contributions are the two set-theoretic systems NF (1936) and ML (1940). 1 These systems were alternatives to the type theory of Principia Mathematica or Zermelo's set theory, and are still being studied by 2 mathematicians. An indirect contribution to the field of logic is his strong resistance to moda110gic. Quine's objectIons to the notions of necessity and analyticity have influenced the development of moda110gic? Quine has had an enormous influence on philosophy of mathematics. When Quine entered philosophy there was a discussion on the foundations of mathematics between the schools of intuitionism, formalism, and conventionalism. Quine soon took issue with Carnap's conventionalism in "Truth by convention,,4 (1936). Quine has never joined one of the other schools, but has added new elements that are the basic ones of the 5 contemporary schools of nominalism, platonism, and structuralism. Quine has long been in the shadow of Benacerraf and Putnam in this field. At the moment there seems to be a renewed interest in Quine's work, and most philosophers explicitly refer to Quine's work.
Author : Abraham Adolf Fraenkel
Publisher : Unknown
Page : 0 pages
File Size : 50,6 Mb
Release : 1976
Category : Set theory
ISBN : 0720422000
Author : William S. Hatcher
Publisher : Elsevier
Page : 330 pages
File Size : 54,6 Mb
Release : 2014-05-09
Category : Mathematics
ISBN : 9781483189635
The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.
Author : Howard DeLong
Publisher : Courier Corporation
Page : 322 pages
File Size : 52,8 Mb
Release : 2012-09-26
Category : Mathematics
ISBN : 9780486139159
This introduction to mathematical logic explores philosophical issues and Gödel's Theorem. Its widespread influence extends to the author of Gödel, Escher, Bach, whose Pulitzer Prize–winning book was inspired by this work.
Author : Manuel Bremer
Publisher : Walter de Gruyter
Page : 125 pages
File Size : 49,8 Mb
Release : 2013-05-02
Category : Philosophy
ISBN : 9783110326109
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?
Author : T. E. Forster
Publisher : Oxford University Press on Demand
Page : 166 pages
File Size : 46,6 Mb
Release : 1995
Category : Mathematics
ISBN : 0198514778
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.
Author : Willard Van O QUINE
Publisher : Harvard University Press
Page : 381 pages
File Size : 40,6 Mb
Release : 2009-06-30
Category : Philosophy
ISBN : 9780674042421
This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted.
Author : John P. Burgess
Publisher : Cambridge University Press
Page : 75 pages
File Size : 52,9 Mb
Release : 2022-02-28
Category : Philosophy
ISBN : 1108986919
Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.
Author : Patrick Suppes
Publisher : Courier Corporation
Page : 265 pages
File Size : 53,8 Mb
Release : 2012-05-04
Category : Mathematics
ISBN : 9780486136875
Geared toward upper-level undergraduates and graduate students, this treatment examines the basic paradoxes and history of set theory and advanced topics such as relations and functions, equipollence, more. 1960 edition.
Author : Willard Van Orman Quine
Publisher : MIT Press
Page : 309 pages
File Size : 54,8 Mb
Release : 2013-01-25
Category : Philosophy
ISBN : 9780262518314
A new edition of Quine's most important work. Willard Van Orman Quine begins this influential work by declaring, "Language is a social art. In acquiring it we have to depend entirely on intersubjectively available cues as to what to say and when." As Patricia Smith Churchland notes in her foreword to this new edition, with Word and Object Quine challenged the tradition of conceptual analysis as a way of advancing knowledge. The book signaled twentieth-century philosophy's turn away from metaphysics and what Churchland calls the "phony precision" of conceptual analysis. In the course of his discussion of meaning and the linguistic mechanisms of objective reference, Quine considers the indeterminacy of translation, brings to light the anomalies and conflicts implicit in our language's referential apparatus, clarifies semantic problems connected with the imputation of existence, and marshals reasons for admitting or repudiating each of various categories of supposed objects. In addition to Churchland's foreword, this edition offers a new preface by Quine's student and colleague Dagfinn Follesdal that describes the never-realized plans for a second edition of Word and Object, in which Quine would offer a more unified treatment of the public nature of meaning, modalities, and propositional attitudes.