Gödel's Incompleteness Theorems

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Gödel's Incompleteness Theorems

Gödel's Incompleteness Theorems

  • Author: Raymond M. Smullyan
  • Publisher: Oxford University Press on Demand
  • ISBN: 0195046722
  • Category: Computers
  • Page: 139
  • View: 2991
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An introduction to the work of the mathematical logician Kurt Godel, which guides the reader through his Theorem of Undecidability and his theories on the completeness of logic, the incompleteness of numbers and the consistency of the axiom of choice.

Fiction and the Incompleteness of History

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Fiction and the Incompleteness of History

Fiction and the Incompleteness of History

Toni Morrison, V.S. Naipaul, and Ben Okri

  • Author: Ying Zhu
  • Publisher: Peter Lang
  • ISBN: 9783039107469
  • Category: Literary Criticism
  • Page: 160
  • View: 8183
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Based on the author's thesis (Doctoral--University of Hong Kong, 2005).

Incompleteness for Higher-Order Arithmetic

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Incompleteness for Higher-Order Arithmetic

Incompleteness for Higher-Order Arithmetic

An Example Based on Harrington’s Principle

  • Author: Yong Cheng
  • Publisher: Springer Nature
  • ISBN: 9811399492
  • Category: Mathematics
  • Page: 122
  • View: 9007
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Gödel's true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements that are similarly unprovable. A lot of research has since been done in this direction, most notably by Harvey Friedman. A lot of examples of concrete incompleteness with real mathematical content have been found to date. This brief contributes to Harvey Friedman's research program on concrete incompleteness for higher-order arithmetic and gives a specific example of concrete mathematical theorems which is expressible in second-order arithmetic but the minimal system in higher-order arithmetic to prove it is fourth-order arithmetic. This book first examines the following foundational question: are all theorems in classic mathematics expressible in second-order arithmetic provable in second-order arithmetic? The author gives a counterexample for this question and isolates this counterexample from the Martin-Harrington Theorem in set theory. It shows that the statement “Harrington's principle implies zero sharp" is not provable in second-order arithmetic. This book further examines what is the minimal system in higher-order arithmetic to prove the theorem “Harrington's principle implies zero sharp" and shows that it is neither provable in second-order arithmetic or third-order arithmetic, but provable in fourth-order arithmetic. The book also examines the large cardinal strength of Harrington's principle and its strengthening over second-order arithmetic and third-order arithmetic.

Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries)

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Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries)

Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries)

  • Author: Rebecca Goldstein
  • Publisher: W. W. Norton & Company
  • ISBN: 0393327604
  • Category: Biography & Autobiography
  • Page: 296
  • View: 4869
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A portrait of the eminent twentieth-century mathematician discusses his theorem of incompleteness, relationships with such contemporaries as Albert Einstein, and untimely death as a result of mental instability and self-starvation.

The Incompleteness Phenomenon

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The Incompleteness Phenomenon

The Incompleteness Phenomenon

  • Author: Martin Goldstern,Haim Judah
  • Publisher: CRC Press
  • ISBN: 1439863539
  • Category: Mathematics
  • Page: 218
  • View: 6473
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This introduction to mathematical logic takes Gödel's incompleteness theorem as a starting point. It goes beyond a standard text book and should interest everyone from mathematicians to philosophers and general readers who wish to understand the foundations and limitations of modern mathematics.

Information, Randomness & Incompleteness

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Information, Randomness & Incompleteness

Information, Randomness & Incompleteness

Papers on Algorithmic Information Theory

  • Author: Gregory J. Chaitin
  • Publisher: World Scientific
  • ISBN: 9789971504809
  • Category: Mathematics
  • Page: 272
  • View: 4419
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The papers gathered in this book were published over a period of more than twenty years in widely scattered journals. They led to the discovery of randomness in arithmetic which was presented in the recently published monograph on ?Algorithmic Information Theory? by the author. There the strongest possible version of G”del's incompleteness theorem, using an information-theoretic approach based on the size of computer programs, was discussed. The present book is intended as a companion volume to the monograph and it will serve as a stimulus for work on complexity, randomness and unpredictability, in physics and biology as well as in metamathematics.

Information-Theoretic Incompleteness

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Information-Theoretic Incompleteness

Information-Theoretic Incompleteness

  • Author: Gregory J. Chaitin
  • Publisher: World Scientific
  • ISBN: 9789810236953
  • Category: Computers
  • Page: 227
  • View: 2012
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In this mathematical autobiography, Gregory Chaitin presents a technical survey of his work and a nontechnical discussion of its significance. The volume is an essential companion to the earlier collection of Chaitin's papers Information, Randomness and Incompleteness, also published by World Scientific.The technical survey contains many new results, including a detailed discussion of LISP program size and new versions of Chaitin's most fundamental information-theoretic incompleteness theorems. The nontechnical part includes the lecture given by Chaitin in G?del's classroom at the University of Vienna, a transcript of a BBC TV interview, and articles from New Scientist, La Recherche, and the Mathematical Intelligencer.

Incompleteness

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Incompleteness

Incompleteness

  • Author: Amy Debellis
  • Publisher: N.A
  • ISBN: 9781945796272
  • Category: Poetry
  • Page: 70
  • View: 9804
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This book of poems concentrates primarily on a relationship rotting from the inside out. It describes certain patterns in the human brain that we all share: the transient sense of identity, the ouroboros loop of denial, and finally, the black flowering of grief. It finishes with the "Eastern ending" of ambiguity, rather than the "Western ending" of a neat conclusion, because our stories are never really over, at least not while we are still alive.

There's Something About Gödel

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There's Something About Gödel

There's Something About Gödel

The Complete Guide to the Incompleteness Theorem

  • Author: Francesco Berto
  • Publisher: John Wiley & Sons
  • ISBN: 1444357611
  • Category: Philosophy
  • Page: 256
  • View: 8512
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Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical style