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| Author | : Rebecca Goldstein |
| Publisher | : W. W. Norton & Company |
| Total Pages | : 299 |
| Release | : 2006-01-31 |
| Genre | : Biography & Autobiography |
| ISBN | : 0393327604 |
"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
| Author | : Raymond M. Smullyan |
| Publisher | : Oxford University Press |
| Total Pages | : 156 |
| Release | : 1992-08-20 |
| Genre | : Mathematics |
| ISBN | : 0195364376 |
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.
| Author | : Peter Smith |
| Publisher | : Cambridge University Press |
| Total Pages | : 376 |
| Release | : 2007-07-26 |
| Genre | : Mathematics |
| ISBN | : 0521857848 |
Peter Smith examines Gödel's Theorems, how they were established and why they matter.
| Author | : Juliette Kennedy |
| Publisher | : Cambridge University Press |
| Total Pages | : 152 |
| Release | : 2022-04-14 |
| Genre | : Philosophy |
| ISBN | : 1108990096 |
This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
| Author | : Lorenz Halbeisen |
| Publisher | : Springer Nature |
| Total Pages | : 236 |
| Release | : 2020-10-16 |
| Genre | : Mathematics |
| ISBN | : 3030522792 |
This book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Gödel’s classical completeness and incompleteness theorems. In particular, the book includes a full proof of Gödel’s second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on the Zermelo’s axioms, containing a presentation of Gödel’s constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers. The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory. Each chapter concludes with a list of exercises.
| Author | : Ernest Nagel |
| Publisher | : Routledge |
| Total Pages | : 109 |
| Release | : 2012-11-12 |
| Genre | : Philosophy |
| ISBN | : 1134953992 |
The first book to present a readable explanation of Godel's theorem to both scholars and non-specialists, this is a gripping combination of science and accessibility, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity.
| Author | : Christopher C. Leary |
| Publisher | : Lulu.com |
| Total Pages | : 382 |
| Release | : 2015 |
| Genre | : Education |
| ISBN | : 1942341075 |
At the intersection of mathematics, computer science, and philosophy, mathematical logic examines the power and limitations of formal mathematical thinking. In this expansion of Leary's user-friendly 1st edition, readers with no previous study in the field are introduced to the basics of model theory, proof theory, and computability theory. The text is designed to be used either in an upper division undergraduate classroom, or for self study. Updating the 1st Edition's treatment of languages, structures, and deductions, leading to rigorous proofs of Gödel's First and Second Incompleteness Theorems, the expanded 2nd Edition includes a new introduction to incompleteness through computability as well as solutions to selected exercises.
| Author | : Torkel Franzén |
| Publisher | : |
| Total Pages | : |
| Release | : 2017-08-09 |
| Genre | : |
| ISBN | : 9781138427266 |
"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
| Author | : Raymond M. Smullyan |
| Publisher | : Princeton University Press |
| Total Pages | : 160 |
| Release | : 1961 |
| Genre | : Mathematics |
| ISBN | : 9780691080475 |
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 | : Kurt Gödel |
| Publisher | : Courier Corporation |
| Total Pages | : 82 |
| Release | : 2012-05-24 |
| Genre | : Mathematics |
| ISBN | : 0486158403 |
First English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.
