Incompleteness For Higher Order Arithmetic

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Incompleteness for Higher-Order Arithmetic

Yong Cheng
Incompleteness for Higher Order Arithmetic Book PDF

Author : Yong Cheng
Publisher : Springer Nature
Page : 122 pages
File Size : 51,5 Mb
Release : 2019-08-30
Category : Mathematics
ISBN : 9789811399497

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Incompleteness for Higher-Order Arithmetic by Yong Cheng Pdf

Gödel's true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements that are similarly unprovable. A lot of research has since been done in this direction, most notably by Harvey Friedman. A lot of examples of concrete incompleteness with real mathematical content have been found to date. This brief contributes to Harvey Friedman's research program on concrete incompleteness for higher-order arithmetic and gives a specific example of concrete mathematical theorems which is expressible in second-order arithmetic but the minimal system in higher-order arithmetic to prove it is fourth-order arithmetic. This book first examines the following foundational question: are all theorems in classic mathematics expressible in second-order arithmetic provable in second-order arithmetic? The author gives a counterexample for this question and isolates this counterexample from the Martin-Harrington Theorem in set theory. It shows that the statement “Harrington's principle implies zero sharp" is not provable in second-order arithmetic. This book further examines what is the minimal system in higher-order arithmetic to prove the theorem “Harrington's principle implies zero sharp" and shows that it is neither provable in second-order arithmetic or third-order arithmetic, but provable in fourth-order arithmetic. The book also examines the large cardinal strength of Harrington's principle and its strengthening over second-order arithmetic and third-order arithmetic.

Incompleteness for Higher-order Arithmetic

Yong Cheng
Incompleteness for Higher order Arithmetic Book PDF

Author : Yong Cheng
Publisher : Unknown
Page : 128 pages
File Size : 54,6 Mb
Release : 2019
Category : Incompleteness theorems
ISBN : 9811399506

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Incompleteness for Higher-order Arithmetic by Yong Cheng Pdf

The book examines the following foundation question: are all theorems in classic mathematics which are expressible in second order arithmetic provable in second order arithmetic? In this book, the author gives a counterexample for this question and isolates this counterexample from Martin-Harrington theorem in set theory. It shows that the statement "Harrington's principle implies zero sharp" is not provable in second order arithmetic. The book also examines what is the minimal system in higher order arithmetic to show that Harrington's principle implies zero sharp and the large cardinal strength of Harrington's principle and its strengthening over second and third order arithmetic.

The Incompleteness Phenomenon

Martin Goldstern,Haim Judah
The Incompleteness Phenomenon Book PDF

Author : Martin Goldstern,Haim Judah
Publisher : CRC Press
Page : 218 pages
File Size : 46,9 Mb
Release : 2018-10-08
Category : Mathematics
ISBN : 9781439863534

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The Incompleteness Phenomenon by Martin Goldstern,Haim Judah Pdf

This introduction to mathematical logic takes Gödel's incompleteness theorem as a starting point. It goes beyond a standard text book and should interest everyone from mathematicians to philosophers and general readers who wish to understand the foundations and limitations of modern mathematics.

Godel's Incompleteness Theorems

Raymond M. Smullyan
Godel s Incompleteness Theorems Book PDF

Author : Raymond M. Smullyan
Publisher : Oxford University Press
Page : 156 pages
File Size : 54,7 Mb
Release : 1992-08-20
Category : Mathematics
ISBN : 9780195364378

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Godel's Incompleteness Theorems by Raymond M. Smullyan Pdf

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

Logical Foundations of Mathematics and Computational Complexity

Pavel Pudlák
Logical Foundations of Mathematics and Computational Complexity Book PDF

Author : Pavel Pudlák
Publisher : Springer Science & Business Media
Page : 699 pages
File Size : 44,9 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9783319001197

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Logical Foundations of Mathematics and Computational Complexity by Pavel Pudlák Pdf

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.

Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective

Mark Burgin
Trilogy Of Numbers And Arithmetic Book 1 History Of Numbers And Arithmetic An Information Perspective Book PDF

Author : Mark Burgin
Publisher : World Scientific
Page : 370 pages
File Size : 43,6 Mb
Release : 2022-04-22
Category : Mathematics
ISBN : 9789811236853

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Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective by Mark Burgin Pdf

The book is the first in the trilogy which will bring you to the fascinating world of numbers and operations with them. Numbers provide information about myriads of things. Together with operations, numbers constitute arithmetic forming in basic intellectual instruments of theoretical and practical activity of people and offering powerful tools for representation, acquisition, transmission, processing, storage, and management of information about the world.The history of numbers and arithmetic is the topic of a variety of books and at the same time, it is extensively presented in many books on the history of mathematics. However, all of them, at best, bring the reader to the end of the 19th century without including the developments in these areas in the 20th century and later. Besides, such books consider and describe only the most popular classes of numbers, such as whole numbers or real numbers. At the same time, a diversity of new classes of numbers and arithmetic were introduced in the 20th century.This book looks into the chronicle of numbers and arithmetic from ancient times all the way to 21st century. It also includes the developments in these areas in the 20th century and later. A unique aspect of this book is its information orientation of the exposition of the history of numbers and arithmetic.

Modal Logic as Metaphysics

Timothy Williamson
Modal Logic as Metaphysics Book PDF

Author : Timothy Williamson
Publisher : OUP Oxford
Page : 480 pages
File Size : 50,7 Mb
Release : 2013-03-28
Category : Philosophy
ISBN : 9780191654763

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Modal Logic as Metaphysics by Timothy Williamson Pdf

Are there such things as merely possible people, who would have lived if our ancestors had acted differently? Are there future people, who have not yet been conceived? Questions like those raise deep issues about both the nature of being and its logical relations with contingency and change. In Modal Logic as Metaphysics, Timothy Williamson argues for positive answers to those questions on the basis of an integrated approach to the issues, applying the technical resources of modal logic to provide structural cores for metaphysical theories. He rejects the search for a metaphysically neutral logic as futile. The book contains detailed historical discussion of how the metaphysical issues emerged in the twentieth century development of quantified modal logic, through the work of such figures as Rudolf Carnap, Ruth Barcan Marcus, Arthur Prior, and Saul Kripke. It proposes higher-order modal logic as a new setting in which to resolve such metaphysical questions scientifically, by the construction of systematic logical theories embodying rival answers and their comparison by normal scientific standards. Williamson provides both a rigorous introduction to the technical background needed to understand metaphysical questions in quantified modal logic and an extended argument for controversial, provocative answers to them. He gives original, precise treatments of topics including the relation between logic and metaphysics, the methodology of theory choice in philosophy, the nature of possible worlds and their role in semantics, plural quantification compared to quantification into predicate position, communication across metaphysical disagreement, and problems for truthmaker theory.

Incompleteness in the Land of Sets

Melvin Fitting
Incompleteness in the Land of Sets Book PDF

Author : Melvin Fitting
Publisher : Unknown
Page : 0 pages
File Size : 46,8 Mb
Release : 2007
Category : Incompleteness theorems
ISBN : 1904987346

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Incompleteness in the Land of Sets by Melvin Fitting Pdf

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.

Essays on Frege's Conception of Truth

Dirk Greimann
Essays on Frege s Conception of Truth Book PDF

Author : Dirk Greimann
Publisher : Rodopi
Page : 258 pages
File Size : 43,5 Mb
Release : 2007
Category : Philosophy
ISBN : 9789042021563

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Essays on Frege's Conception of Truth by Dirk Greimann Pdf

In his writings on the foundations of logic, Gottlob Frege, the father of modern logic, sketched a conception of truth that focuses on the following questions: What is the sense of the word "true"? Is truth a definable concept or a primitive one? What are the kinds of things of which truth is predicated? What is the role of the concept of truth in judgment, assertion and recognition? What is the logical category of truth? What is the significance of the concept of truth for science in general and for logic in particular? The present volume is dedicated to the interpretation, reconstruction and critical assessment of Frege's conception of truth. It is of interest to all those working on Frege, the history of logic and semantics, or theories of truth. The volume brings together nine original papers whose authors are all widely known to Frege scholars. The main topics are: the role of the concept of truth in Frege's system, the nature of the truth-values, the logical category of truth, the relationship between truth and judgment, and the conception of the truth-bearers.

Foundations without Foundationalism

Stewart Shapiro
Foundations without Foundationalism Book PDF

Author : Stewart Shapiro
Publisher : Clarendon Press
Page : 302 pages
File Size : 47,8 Mb
Release : 1991-09-19
Category : Mathematics
ISBN : 9780191524011

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Foundations without Foundationalism by Stewart Shapiro Pdf

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject.

Incompleteness and Computability

Richard Zach
Incompleteness and Computability Book PDF

Author : Richard Zach
Publisher : Unknown
Page : 281 pages
File Size : 41,7 Mb
Release : 2019-11-09
Category : Electronic
ISBN : 1077323395

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Incompleteness and Computability by Richard Zach Pdf

This book is an introduction to metamathematics and Gödel's theorems. It covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus. It is based on the Open Logic Project, and available for free download at ic.openlogicproject.org.

Reading Frege's Grundgesetze

Richard G. Heck
Reading Frege s Grundgesetze Book PDF

Author : Richard G. Heck
Publisher : Oxford University Press on Demand
Page : 315 pages
File Size : 44,9 Mb
Release : 2012-11-29
Category : Mathematics
ISBN : 9780199233700

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Reading Frege's Grundgesetze by Richard G. Heck Pdf

Readership: Scholars and advanced students of philosophy of logic, philosophy of mathematics, and history of analytic philosophy

Gödel's Theorem

Torkel Franzén
G del s Theorem Book PDF

Author : Torkel Franzén
Publisher : Unknown
Page : 128 pages
File Size : 41,7 Mb
Release : 2017-08-09
Category : Electronic
ISBN : 1138427268

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Gödel's Theorem by Torkel Franzén Pdf

"Among the many expositions of G�del's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franz�gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature." --- John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt G�del

Kurt Gödel and the Foundations of Mathematics

Matthias Baaz,Christos H. Papadimitriou,Hilary W. Putnam,Dana S. Scott,Charles L. Harper, Jr
Kurt G del and the Foundations of Mathematics Book PDF

Author : Matthias Baaz,Christos H. Papadimitriou,Hilary W. Putnam,Dana S. Scott,Charles L. Harper, Jr
Publisher : Cambridge University Press
Page : 541 pages
File Size : 54,6 Mb
Release : 2011-06-06
Category : Mathematics
ISBN : 9781139498432

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Kurt Gödel and the Foundations of Mathematics by Matthias Baaz,Christos H. Papadimitriou,Hilary W. Putnam,Dana S. Scott,Charles L. Harper, Jr Pdf

This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Logic, Language, and Mathematics

Alexander Miller
Logic Language and Mathematics Book PDF

Author : Alexander Miller
Publisher : Oxford University Press, USA
Page : 465 pages
File Size : 47,6 Mb
Release : 2020-03-19
Category : Mathematics
ISBN : 9780199278343

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Logic, Language, and Mathematics by Alexander Miller Pdf

Crispin Wright is widely recognised as one of the most important and influential analytic philosophers of the twentieth and twenty-first centuries. This volume is a collective exploration of the major themes of his work in philosophy of language, philosophical logic, and philosophy of mathematics. It comprises specially written chapters by a group of internationally renowned thinkers, as well as four substantial responses from Wright. In these thematically organized replies, Wright summarizes his life's work and responds to the contributory essays collected in this book. In bringing together such scholarship, the present volume testifies to both the enormous interest in Wright's thought and the continued relevance of Wright's seminal contributions in analytic philosophy for present-day debates;