Godel's Proof
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Godel's Proof

Ernest Nagel, James R. Newman

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eBook - ePub

Godel's Proof

Ernest Nagel, James R. Newman

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About This Book

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.

Marking the 50th anniversary of the original publication of Gödel's Proof, New York University Press is proud to publish this special anniversary edition of one of its bestselling and most frequently translated 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.

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Information

Publisher
NYU Press
Year
2001
ISBN
9780814758014

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