Gödel's Proof

Ernest Nagel, James R Newman, Douglas R. Hofstadter, James R. Newman

• Publisher: NYU Press
• Year: 2001
• Print ISBN: 9780814758373
• eBook ISBN: 9780814758014
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

An accessible explanation of Kurt Gödel's groundbreaking work in mathematical logic

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.

Index

absolute proofs of consistency, 25–36, 45–56, 109, 120–124

antinomies, 22–24, 60–63, 92–93

Archimedes, 3

Aristotle, 39, 40

arithmetic, formalized: incompleteness of, 4–5, 93–95, 102–106

consistency of, 57–58, 67, 94, 104–108

see also Principia Mathematica

arithmetization: of algebra and calculus, 42

of formalized arithmetic, 68–78

of metamathematics, 80–87

axiomatic method, 2–4

limitations of, 4–5, 56, 58–59, 109–112

axioms: meaning of, 2–4

of the sentential calculus, 48–56

of Peano, 114

of PM, 48–50, 93–94, 103–104

Bolyai, Janos, 9

Boole, George, 40–41

calculating machines and human intelligence, 111–112

calculus of signs, 25–36, 45

Cantor, Georg, 22, 107n

chess and meta-chess, 34–36

class: notion of, 15

mathematical theory of, 22–23

completeness, 55–56, 102

computers, see calculating machines

consistency: formally characterized, 30, 33–34, 50–51, 104

problem of, 7–24

and truth, 13–19

of Euclidean geometry, 13–14, 17–20

of non-Euclidean geometry, 14, 16–19

relative proofs of, 14–20

absolute proofs of, 25–26, 32–36, 45–55, 57, 109

of the sentential calculus, 45–55, 120–124

of formalized arithmetic (PM), 5, 30, 33–34, 58, 67, 94, 99–100n, 104–108, 109

Correspondence Lemma, 72–73, 83, 85–87, 89, 90n

creative reason, powers of, 113

defined symbols in PM, 76n

dem (x, z), 84–85

dem’

(x, z), 85, 103

Dem (x, z), 85–87

demonstration, definition of, 46

Descartes, René, 5

descriptive predicates, 11

duality of points and lines, 64–65

elementary signs, 46, 69

elliptic geometry, 14, 16–19

essential incompletenes...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Foreword to the New Edition by Douglas R. Hofstadter
• Acknowledgments
• I Introduction
• II The Problem of Consistency
• III Absolute Proofs of Consistency
• IV The Systematic Codification of Formal Logic
• V An Example of a Successful Absolute Proof of Consistency
• VI The Idea of Mapping and Its Use in Mathematics
• VII Gödel’s Proofs
• VIII Concluding Reflections
• Appendix: Notes
• Brief Bibliography
• Index
• footnotes