Search Results: incompleteness-in-the-land-of-sets

Incompleteness in the Land of Sets

Author: Melvin Fitting

Publisher: N.A

ISBN: 9781904987345

Category: Mathematics

Page: 142

View: 9092

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 Godel, 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. Godel 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 Godel's Second Incompleteness Theorem. Exercises are scattered throughout.

First-Order Logic and Automated Theorem Proving

Author: Melvin Fitting

Publisher: Springer Science & Business Media

ISBN: 1468403575

Category: Mathematics

Page: 242

View: 6771

There are many kinds of books on formal logic. Some have philosophers as their intended audience, some mathematicians, some computer scientists. Although there is a common core to all such books they will be very dif ferent in emphasis, methods, and even appearance. This book is intended for computer scientists. But even this is not precise. Within computer sci ence formal logic turns up in a number of areas, from program verification to logic programming to artificial intelligence. This book is intended for computer scientists interested in automated theorem proving in classical logic. To be more precise yet, it is essentially a theoretical treatment, not a how-to book, although how-to issues are not neglected. This does not mean, of course, that the book will be of no interest to philosophers or mathematicians. It does contain a thorough presentation of formal logic and many proof techniques, and as such it contains all the material one would expect to find in a course in formal logic covering completeness but not incompleteness issues. The first item to be addressed is, what are we talking about and why are we interested in it. We are primarily talking about truth as used in mathematical discourse, and our interest in it is, or should be, self-evident. Truth is a semantic concept, so we begin with models and their properties. These are used to define our subject.

Raymond Smullyan on Self Reference

Author: Melvin Fitting,Brian Rayman

Publisher: Springer

ISBN: 3319687328

Category: Philosophy

Page: 195

View: 4416

This book collects, for the first time in one volume, contributions honoring Professor Raymond Smullyan’s work on self-reference. It serves not only as a tribute to one of the great thinkers in logic, but also as a celebration of self-reference in general, to be enjoyed by all lovers of this field. Raymond Smullyan, mathematician, philosopher, musician and inventor of logic puzzles, made a lasting impact on the study of mathematical logic; accordingly, this book spans the many personalities through which Professor Smullyan operated, offering extensions and re-evaluations of his academic work on self-reference, applying self-referential logic to art and nature, and lastly, offering new puzzles designed to communicate otherwise esoteric concepts in mathematical logic, in the manner for which Professor Smullyan was so well known. This book is suitable for students, scholars and logicians who are interested in learning more about Raymond Smullyan's work and life.

Infinity and the Mind

The Science and Philosophy of the Infinite

Author: Rudy Rucker

Publisher: Princeton University Press

ISBN: 1400849047

Category: Mathematics

Page: 368

View: 731

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.

A Book of Set Theory

Author: Charles C Pinter

Publisher: Courier Corporation

ISBN: 0486497089

Category: Mathematics

Page: 256

View: 5555

"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"--

Rationalist Spirituality

An exploration of the meaning of life and existence informed by logic and science

Author: Bernardo Kastrup

Publisher: John Hunt Publishing

ISBN: 1846947359

Category: Philosophy

Page: 125

View: 3520

Why does the universe exist and what are you supposed to do in it? This question has been addressed by religions since time immemorial, but popular answers often fail to account for obvious aspects of reality. Indeed, if God knows everything, why do we need to learn through pain and suffering? If God is omnipotent, why are we needed to do good? If the universe is fundamentally good, why are wars, crime, and injustice all around us? In modern society, orthodox science takes the rational high-ground and tackles these contradictions by denying the very need for, and the existence of, meaning. Indeed, many of us implicitly accept the notion that rationality somehow contradicts spirituality. That is a modern human tragedy, not only for its insidiousness, but for the fact that it is simply not true. In this book, the author constructs a coherent and logical argument for the meaning of existence, informed by science itself. A framework is laid out wherein all aspects of human existence have a logical, coh

Set Theory and the Continuum Problem

Author: Raymond M. Smullyan,Melvin Fitting

Publisher: N.A

ISBN: 9780486474847

Category: Mathematics

Page: 315

View: 8982

A lucid, elegant, and complete survey of set theory, this three-part treatment explores axiomatic set theory, the consistency of the continuum hypothesis, and forcing and independence results. 1996 edition.

First-Order Modal Logic

Author: M. Fitting,Richard L. Mendelsohn

Publisher: Springer Science & Business Media

ISBN: 9401152926

Category: Philosophy

Page: 292

View: 6619

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

Bloodlands

Europe Between Hitler and Stalin

Author: Timothy Snyder

Publisher: Basic Books

ISBN: 0465032974

Category: History

Page: 544

View: 8971

From the bestselling author of On Tyranny, the definitive history of Hitler's and Stalin's wars against the civilians of Europe in World War Two Americans call the Second World War "The Good War."But before it even began, America's wartime ally Josef Stalin had killed millions of his own citizens--and kept killing them during and after the war. Before Hitler was finally defeated, he had murdered six million Jews and nearly as many other Europeans. At war's end, both the German and the Soviet killing sites fell behind the iron curtain, leaving the history of mass killing in darkness. Bloodlands is a new kind of European history, presenting the mass murders committed by the Nazi and Stalinist regimes as two aspects of a single history, in the time and place where they occurred: between Germany and Russia, when Hitler and Stalin both held power. Assiduously researched, deeply humane, and utterly definitive, Bloodlands will be required reading for anyone seeking to understand the central tragedy of modern history. Bloodlands won twelve awards including the Emerson Prize in the Humanities, a Literature Award from the American Academy of Arts and Letters, the Leipzig Award for European Understanding, and the Hannah Arendt Prize in Political Thought. It has been translated into more than thirty languages, was named to twelve book-of-the-year lists, and was a bestseller in six countries.

In the Light of What We Know

A Novel

Author: Zia Haider Rahman

Publisher: Farrar, Straus and Giroux

ISBN: 0374710082

Category: Fiction

Page: 512

View: 1025

A bold, epic debut novel set during the war and financial crisis that defined the beginning of our century One September morning in 2008, an investment banker approaching forty, his career in collapse and his marriage unraveling, receives a surprise visitor at his West London townhouse. In the disheveled figure of a South Asian male carrying a backpack, the banker recognizes a long-lost friend, a mathematics prodigy who disappeared years earlier under mysterious circumstances. The friend has resurfaced to make a confession of unsettling power. In the Light of What We Know takes us on a journey of exhilarating scope--from Kabul to London, New York, Islamabad, Oxford, and Princeton--and explores the great questions of love, belonging, science, and war. It is an age-old story: the friendship of two men and the betrayal of one by the other. The visitor, a man desperate to climb clear of his wrong beginnings, seeks atonement; and the narrator sets out to tell his friend's story but finds himself at the limits of what he can know about the world--and, ultimately, himself. Set against the breaking of nations and beneath the clouds of economic crisis, this surprisingly tender novel chronicles the lives of people carrying unshakable legacies of class and culture as they struggle to tame their futures. In an extraordinary feat of imagination, Zia Haider Rahman has telescoped the great upheavals of our young century into a novel of rare intimacy and power.

Down These Green Streets

Irish Crime Writing in the 21st Century

Author: Declan Burke

Publisher: Liberties Press

ISBN: 1909718041

Category: Fiction

Page: 368

View: 5745

This book suggests crime fiction is now the most relevant and valid form of writing which can deal with modern Ireland in terms of the post-'Troubles' landscape and post-Celtic Tiger economic boom. The book takes a chapter by chapter approach with each chapter and author discussing a different facet of Irish crime writing for example, Declan Hughes discusses the influence of American culture on Irish crime writing and Tana French reflects on crime fiction and the post-Celtic Tiger Irish identity. This publication is aimed at both the academic and general reader.

Forever Undecided

Author: Raymond M. Smullyan

Publisher: Knopf

ISBN: 0307962466

Category: Mathematics

Page: 257

View: 6815

Forever Undecided is the most challenging yet of Raymond Smullyan’s puzzle collections. It is, at the same time, an introduction—ingenious, instructive, entertaining—to Gödel’s famous theorems. With all the wit and charm that have delighted readers of his previous books, Smullyan transports us once again to that magical island where knights always tell the truth and knaves always lie. Here we meet a new and amazing array of characters, visitors to the island, seeking to determine the natives’ identities. Among them: the census-taker McGregor; a philosophical-logician in search of his flighty bird-wife, Oona; and a regiment of Reasoners (timid ones, normal ones, conceited, modest, and peculiar ones) armed with the rules of propositional logic (if X is true, then so is Y). By following the Reasoners through brain-tingling exercises and adventures—including journeys into the “other possible worlds” of Kripke semantics—even the most illogical of us come to understand Gödel’s two great theorems on incompleteness and undecidability, some of their philosophical and mathematical implications, and why we, like Gödel himself, must remain Forever Undecided!

Logic and Structure

Author: Dirk van Dalen

Publisher: Springer Science & Business Media

ISBN: 3662029626

Category: Mathematics

Page: 220

View: 7162

New corrected printing of a well-established text on logic at the introductory level.

Shadows of the Mind

A Search for the Missing Science of Consciousness

Author: Roger Penrose

Publisher: Oxford University Press, USA

ISBN: 9780195106466

Category: Computers

Page: 457

View: 6958

Presenting a look at the human mind's capacity while criticizing artificial intelligence, the author makes suggestions about classical and quantum physics and the role of microtubules

The Foundations of Mathematics

Author: Kenneth Kunen

Publisher: N.A

ISBN: 9781904987147

Category: Mathematics

Page: 251

View: 3701

Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth.

The Selected Works of T. S. Spivet

A Novel

Author: Reif Larsen

Publisher: Penguin

ISBN: 0698148231

Category: Fiction

Page: 400

View: 4273

Discover The Selected Works of T.S. Spivet for iPad. A brilliant, boundary-leaping debut novel tracing twelve-year-old genius map maker T.S. Spivet's attempts to understand the ways of the world When twelve-year-old genius cartographer T.S. Spivet receives an unexpected phone call from the Smithsonian announcing he has won the prestigious Baird Award, life as normal-if you consider mapping family dinner table conversation normal-is interrupted and a wild cross-country adventure begins, taking T.S. from his family ranch just north of Divide, Montana, to the museum's hallowed halls. T.S. sets out alone, leaving before dawn with a plan to hop a freight train and hobo east. Once aboard, his adventures step into high gear and he meticulously maps, charts, and illustrates his exploits, documenting mythical wormholes in the Midwest, the urban phenomenon of "rims," and the pleasures of McDonald's, among other things. We come to see the world through T.S.'s eyes and in his thorough investigation of the outside world he also reveals himself. As he travels away from the ranch and his family we learn how the journey also brings him closer to home. A secret family history found within his luggage tells the story of T.S.'s ancestors and their long-ago passage west, offering profound insight into the family he left behind and his role within it. As T.S. reads he discovers the sometimes shadowy boundary between fact and fiction and realizes that, for all his analytical rigor, the world around him is a mystery. All that he has learned is tested when he arrives at the capital to claim his prize and is welcomed into science's inner circle. For all its shine, fame seems more highly valued than ideas in this new world and friends are hard to find. T.S.'s trip begins at the Copper Top Ranch and the last known place he stands is Washington, D.C., but his journey's movement is far harder to track: How do you map the delicate lessons learned about family and self? How do you depict how it feels to first venture out on your own? Is there a definitive way to communicate the ebbs and tides of heartbreak, loss, loneliness, love? These are the questions that strike at the core of this very special debut. Now a major motion picture directed by Jean-Pierre Jeunet and starring Kyle Catlett and Helena Bonham Carter. From the Hardcover edition.

A Discourse and Register Analysis of the Prophetic Book of Joel

Author: Colin Toffelmire

Publisher: BRILL

ISBN: 9004325077

Category: Religion

Page: 242

View: 3342

In A Discourse and Register Analysis of the Prophetic Book of Joel, Colin M. Toffelmire presents a thorough analysis of the text of Joel from the perspective of Systemic Functional Linguistics.

The Bulletin of Symbolic Logic

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Electronic journals

Page: N.A

View: 8857

Subsystems of Second Order Arithmetic

Author: Stephen George Simpson

Publisher: Cambridge University Press

ISBN: 052188439X

Category: Mathematics

Page: 444

View: 9376

This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.

A Mathematical Introduction to Logic

Author: Herbert Enderton,Herbert B. Enderton

Publisher: Elsevier

ISBN: 0080496466

Category: Mathematics

Page: 317

View: 5625

A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional coverage of introductory material such as sets. * Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses. * Reduced mathematical rigour to fit the needs of undergraduate students

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