Author: Charles Allan Green
Publisher:
ISBN:
Category : Logic, Symbolic and mathematical
Languages : en
Pages : 114

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Author: Victor Christianto
Publisher: Infinite Study
ISBN:
Category : Mathematics
Languages : en
Pages : 5

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Book Description
Physics thrives on precision, but paradoxes in set theory reveal limitations in our understanding of well-defined boundaries. Neutrosophic logic, challenging the excluded middle principle, introduces the concept of "betweenness" and partial belonging. This article explore among other things several possible avenues to resolve set theory paradoxism, including potential applications of neutrosophic logic in cosmology and particles, from set boundary, to the hypothetical "cosmosphere" boundary, to mixed fermion-boson condensate hypothesis. Embracing indeterminacy and fuzzy boundaries paves the way for a more holistic understanding of the universe's complexity.

Author: Russell Zelany Christensen
Publisher:
ISBN: 9780473206482
Category : Liar paradox
Languages : en
Pages : 186

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Author: Melvin Fitting
Publisher:
ISBN: 9781904987345
Category : Incompleteness theorems
Languages : en
Pages : 0

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Book Description
Russell's paradox arises when we consider those sets that do not belong to themselves. The collection of such sets cannot constitute a set. Step back a bit. Logical formulas define sets (in a standard model). Formulas, being mathematical objects, can be thought of as sets themselves-mathematics reduces to set theory. Consider those formulas that do not belong to the set they define. The collection of such formulas is not definable by a formula, by the same argument that Russell used. This quickly gives Tarski's result on the undefinability of truth. Variations on the same idea yield the famous results of Gödel, Church, Rosser, and Post. This book gives a full presentation of the basic incompleteness and undecidability theorems of mathematical logic in the framework of set theory. Corresponding results for arithmetic follow easily, and are also given. Gödel numbering is generally avoided, except when an explicit connection is made between set theory and arithmetic. The book assumes little technical background from the reader. One needs mathematical ability, a general familiarity with formal logic, and an understanding of the completeness theorem, though not its proof. All else is developed and formally proved, from Tarski's Theorem to Gödel's Second Incompleteness Theorem. Exercises are scattered throughout.

Author: Stewart Shapiro
Publisher: Routledge
ISBN: 1351886665
Category : Philosophy
Languages : en
Pages : 352

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Book Description
The International research Library of Philosophy collects in book form a wide range of important and influential essays in philosophy, drawn predominantly from English-language journals. Each volume in the library deals with a field of enquiry which has received significant attention in philosophy in the last 25 years and is edited by a philosopher noted in that field.

Author: Scott Soames
Publisher: Princeton University Press
ISBN: 1400850452
Category : Philosophy
Languages : en
Pages : 676

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Book Description
This is the first of five volumes of a definitive history of analytic philosophy from the invention of modern logic in 1879 to the end of the twentieth century. Scott Soames, a leading philosopher of language and historian of analytic philosophy, provides the fullest and most detailed account of the analytic tradition yet published, one that is unmatched in its chronological range, topics covered, and depth of treatment. Focusing on the major milestones and distinguishing them from the dead ends, Soames gives a seminal account of where the analytic tradition has been and where it appears to be heading. Volume 1 examines the initial phase of the analytic tradition through the major contributions of three of its four founding giants—Gottlob Frege, Bertrand Russell, and G. E. Moore. Soames describes and analyzes their work in logic, the philosophy of mathematics, epistemology, metaphysics, ethics, and the philosophy of language. He explains how by about 1920 their efforts had made logic, language, and mathematics central to philosophy in an unprecedented way. But although logic, language, and mathematics were now seen as powerful tools to attain traditional ends, they did not yet define philosophy. As volume 1 comes to a close, that was all about to change with the advent of the fourth founding giant, Ludwig Wittgenstein, and the 1922 English publication of his Tractatus, which ushered in a "linguistic turn" in philosophy that was to last for decades.

Author: E. Brian Davies
Publisher: OUP Oxford
ISBN: 0191527432
Category : Science
Languages : en
Pages : 308

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Book Description
How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? In this wide-ranging book, Brian Davies discusses the basis for scientists' claims to knowledge about the world. He looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. He rejects the claim that all scientific knowledge is provisional, by citing examples from chemistry, biology and geology. A major feature of the book is its defence of the view that mathematics was invented rather than discovered. While experience has shown that disentangling knowledge from opinion and aspiration is a hard task, this book provides a clear guide to the difficulties. Full of illuminating examples and quotations, and with a scope ranging from psychology and evolution to quantum theory and mathematics, this book brings alive issues at the heart of all science.

Author:
Publisher: Elsevier
ISBN: 0080930662
Category : Mathematics
Languages : en
Pages : 878

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Book Description
Set theory is an autonomous and sophisticated field of mathematics that is extremely successful at analyzing mathematical propositions and gauging their consistency strength. It is as a field of mathematics that both proceeds with its own internal questions and is capable of contextualizing over a broad range, which makes set theory an intriguing and highly distinctive subject. This handbook covers the rich history of scientific turning points in set theory, providing fresh insights and points of view. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in mathematics, the history of philosophy, and any discipline such as computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration Serves as a singular contribution to the intellectual history of the 20th century Contains the latest scholarly discoveries and interpretative insights

Author: D.C. Goldrei
Publisher: Routledge
ISBN: 1351460617
Category : Mathematics
Languages : en
Pages : 296

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Book Description
Designed for undergraduate students of set theory, Classic Set Theory presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. This includes:The definition of the real numbers in terms of rational numbers and ultimately in terms of natural numbersDefining natural numbers in terms of setsThe potential paradoxes in set theoryThe Zermelo-Fraenkel axioms for set theoryThe axiom of choiceThe arithmetic of ordered setsCantor's two sorts of transfinite number - cardinals and ordinals - and the arithmetic of these.The book is designed for students studying on their own, without access to lecturers and other reading, along the lines of the internationally renowned courses produced by the Open University. There are thus a large number of exercises within the main body of the text designed to help students engage with the subject, many of which have full teaching solutions. In addition, there are a number of exercises without answers so students studying under the guidance of a tutor may be assessed.Classic Set Theory gives students sufficient grounding in a rigorous approach to the revolutionary results of set theory as well as pleasure in being able to tackle significant problems that arise from the theory.

Author: Herbert B. Enderton
Publisher: Academic Press
ISBN: 0080570429
Category : Mathematics
Languages : en
Pages : 294

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Book Description
This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.