Gödel's proof

"In 1931 Kurt Godel disrupted some of the fundamental assumptions underlying mathematics and logic with the publication of his revolutionary paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." Ironically, few mathematicians of the time were able to understand the young scholar's complex proof, and the full importance of this work was largely overlooked for many years. Godel was at last recognized by his peers and presented with the first Albert Einstein Award in 1951 for achievement in the natural sciences - the highest honor of its kind in the United States. The award committee, which included Albert Einstein and J. Robert Oppenheimer, described his work as "one of the greatest contributions to the sciences in recent times."" "In Godel's Proof, Ernest Nagel and James Newman provide a readable and non-technical explanation for both scholars and non-specialists of the main ideas and broad implications of Godel's discovery. First published in 1958 and in print continuously in ten languages, this highly popular, seminal work offers every educated person with an interest in mathematics, logic, and philosophy the opportunity to understand a previously difficult and inaccessible subject."--BOOK JACKET. Read more... The problem of consistency -- Absolute proofs of consistency -- The systematic codification of formal logic -- An example of a successful absolute proof of consistency -- The idea of mapping and its use in mathematics -- Gödel's proofs -- A Gödel numbering -- The arithmetization of meta-mathematics -- The heart of Gödel's argument -- Concluding reflections -- Appendix: Notes -- Brief bibliography -- Index In 1931 Kurt Godel Disrupted Some Of The Fundamental Assumptions Underlying Mathematics And Logic With The Publication Of His Revolutionary Paper, On Formally Undecidable Propositions Of Principia Mathematica And Related Systems. Ironically, Few Mathematicians Of The Time Were Able To Understand The Young Scholar's Complex Proof, And The Full Importance Of This Work Was Largely Overlooked For Many Years. Godel Was At Last Recognized By His Peers And Presented With The First Albert Einstein Award In 1951 For Achievement In The Natural Sciences - The Highest Honor Of Its Kind In The United States. The Award Committee, Which Included Albert Einstein And J. Robert Oppenheimer, Described His Work As One Of The Greatest Contributions To The Sciences In Recent Times. In Godel's Proof, Ernest Nagel And James Newman Provide A Readable And Non-technical Explanation For Both Scholars And Non-specialists Of The Main Ideas And Broad Implications Of Godel's Discovery. First Published In 1958 And In Print Continuously In Ten Languages, This Highly Popular, Seminal Work Offers Every Educated Person With An Interest In Mathematics, Logic, And Philosophy The Opportunity To Understand A Previously Difficult And Inaccessible Subject.--jacket. The Problem Of Consistency -- Absolute Proofs Of Consistency -- The Systematic Codification Of Formal Logic -- An Example Of A Successful Absolute Proof Of Consistency -- The Idea Of Mapping And Its Use In Mathematics -- Gödel's Proofs -- A Gödel Numbering -- The Arithmetization Of Meta-mathematics -- The Heart Of Gödel's Argument -- Concluding Reflections -- Appendix: Notes -- Brief Bibliography -- Index. By Ernest Nagel And James R. Newman ; Edited And With A Foreward By Douglas R. Hofstadter. Includes Bibliographical References (p. 125) And Index. An accessible explanation of Kurt Gödel's groundbreaking work in mathematical logicIn 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 description: "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." In 1931 Kurt Godel disrupted some of the fundamental assumptions underlying mathematics and logic with the publication of his revolutionary paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems". Ironically, few mathematicians of the time were able to understand the young scholar's complex proof, and the full importance of this work was largely overlooked for many years. Godel was at last recognized by his peers and presented with the first Albert Einstein Award in 1951 for achievement in the natural sciences -- the highest honor of its kind in the United States. The award committee, which included Albert Einstein and J. Robert Oppenheimer, described his work in as "one of the greatest contributions to the sciences in recent times".In Godel's Proof Ernest Nagel and James Newman provide a readable and non-technical explanation for both scholars and non-specialists of the main ideas and broad implications of Godel's theory. First published in 1958 and in print continuously since then in 10 languages, this highly popular, seminal work offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. In 1931 Kurt Godel 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. Godel 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 Godel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. 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. Machine generated contents note: Foreword to the New Edition by Douglas R Hofstadter ix Acknowledgments xxiii i Introduction I II The Problem of Consistency 7 inI Absolute Proofs of Consistency 25 iv The Systematic Codification of Formal Logic 37 v An Example of a Successful Absolute Proof of Consistency 45 vI The Idea of Mapping and Its Use in Mathematics 57 vii G6del'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 Index 127.