In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."
However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.
New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.
|Publisher:||New York University Press|
|Product dimensions:||5.00(w) x 8.00(h) x 0.60(d)|
About the Author
Ernest Nagel was John Dewey Professor of Philosophy at Columbia University.
James R. Newman was the author of What is Science.
Douglas R. Hofstadter is College of Arts and Sciences Professor of computer science and cognitive science at Indiana University and author of the Pulitzer-prize winning Gödel, Escher, Bach: An Eternal Golden Braid.
Table of Contents
Foreword to the New Edition by Douglas R. Hofstadter ix
Acknowledgments xxiii i Introduction 1
ii The Problem of Consistency 7
iii Absolute Proofs of Consistency 25
iv The Systematic Codification of Formal Logic 37
v An Example of a Successful Absolute Proof of
vi The Idea of Mapping and Its Use in Mathematics 57
vii Godel's Proofs 68
a Godel numbering 68
b The arithmetization of meta-mathematics 80
c The heart of Godel's argument 92
viii Concluding Reflections 109
Appendix: Notes 114
Brief Bibliography 125
What People are Saying About This
"A little masterpiece of exegesis."
"An excellent non-technical account of the substance of Gödel's celebrated paper."
-Bulletin of the American Mathematical Society
Most Helpful Customer Reviews
Great introductory book on Godel´s incompletness theorem. Starts with a clear explanation about how simple axioms become theorems and some of the problems associated with consitency. Next it will guide you through the requirements to grasp Godel´s proof and at the end it will provide a clear explanation on the subject. It will even explain what mathematical formality is all about. Don´t worry about your mathematical background the authors do a great job on explaining everything as simple as possible. Once I started reading I couldn´t stop. I recommend reading it before the formal paper written by Godel itself.
This is a non-formal, though still rigorous, presentation of the argument of Gödel's famous demonstration that will be accessible to anyone familiar with the basics of mathematical proof, logic, and number theory. By the end of the book, I acutally had the outline of Gödel's tricky self-referential argument all in my head at once, and though it faded quickly, I feel confident I could resurrect it with another reading. Nagel's description of the significance of the proof, as opposed to its mechanics, is less thorough, but that's a quibble. This slim book is a truly impressive feat of exposition.
Simply excellent. You will understand this "piece of jewell" (which is not a minor stuff concerning this theorem..)!!!
This book does the best job of explaining a fundamentally opaque subject matter clearly and concisely to the lay reader, especially with the new footnotes added in by Douglas Hofstadter in this editione. i highly recommend this title to those interested in the fundamentals of mathematics, logic, or computer science.