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Author | : Yong Cheng |
Publisher | : Springer Nature |
Total Pages | : 122 |
Release | : 2019-08-30 |
Genre | : Mathematics |
ISBN | : 9811399492 |
Download Incompleteness for Higher-Order Arithmetic Book in PDF, ePub and Kindle
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.
Author | : Yong Cheng |
Publisher | : |
Total Pages | : |
Release | : 2019 |
Genre | : Incompleteness theorems |
ISBN | : 9789811399510 |
Download Incompleteness for Higher-order Arithmetic Book in PDF, ePub and Kindle
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.
Author | : Raymond M. Smullyan |
Publisher | : Princeton University Press |
Total Pages | : 160 |
Release | : 1961 |
Genre | : Mathematics |
ISBN | : 9780691080475 |
Download Theory of Formal Systems Book in PDF, ePub and Kindle
This book serves both as a completely self-contained introduction and as an exposition of new results in the field of recursive function theory and its application to formal systems.
Author | : Alfred North Whitehead |
Publisher | : |
Total Pages | : 688 |
Release | : 1910 |
Genre | : Logic, Symbolic and mathematical |
ISBN | : |
Download Principia Mathematica Book in PDF, ePub and Kindle
Author | : Torkel Franzén |
Publisher | : CRC Press |
Total Pages | : 182 |
Release | : 2005-06-06 |
Genre | : Mathematics |
ISBN | : 1439876924 |
Download Gödel's Theorem Book in PDF, ePub and Kindle
"Among the many expositions of Gödel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén 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 Gödel
Author | : Peter Smith |
Publisher | : Cambridge University Press |
Total Pages | : 376 |
Release | : 2007-07-26 |
Genre | : Mathematics |
ISBN | : 1139465937 |
Download An Introduction to Gödel's Theorems Book in PDF, ePub and Kindle
In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.
Author | : Rebecca Goldstein |
Publisher | : W. W. Norton & Company |
Total Pages | : 299 |
Release | : 2006-01-31 |
Genre | : Biography & Autobiography |
ISBN | : 0393327604 |
Download Incompleteness Book in PDF, ePub and Kindle
"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
Author | : Stewart Shapiro |
Publisher | : Clarendon Press |
Total Pages | : 302 |
Release | : 1991-09-19 |
Genre | : Mathematics |
ISBN | : 0191524018 |
Download Foundations without Foundationalism Book in PDF, ePub and Kindle
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.
Author | : Peter B. Andrews |
Publisher | : Springer Science & Business Media |
Total Pages | : 404 |
Release | : 2013-04-17 |
Genre | : Mathematics |
ISBN | : 9401599343 |
Download An Introduction to Mathematical Logic and Type Theory Book in PDF, ePub and Kindle
In case you are considering to adopt this book for courses with over 50 students, please contact [email protected] for more information. This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.
Author | : Martin Goldstern |
Publisher | : CRC Press |
Total Pages | : 218 |
Release | : 2018-10-08 |
Genre | : Mathematics |
ISBN | : 1439863539 |
Download The Incompleteness Phenomenon Book in PDF, ePub and Kindle
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.