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Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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The Impact of the Incompleteness Theorems On Mathematics
, 2006
"... In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits an ..."
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In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about what he calls The Paradox of Irrelevance: “There are many math problems that have achieved the cachet of tremendous significance, e.g., Fermat, fourcolor, Kepler’s packing, Gödel, etc. Of Fermat, I have read: ‘the most famous math problem of all time’. Of Gödel, I have read: ‘the most mathematically significant achievement of the 20th century’. … Yet, these problems have engaged the attention of relatively few research mathematicians—even in pure math. ” What accounts for this disconnect between fame and relevance? Before going into the question for Gödel’s theorems, it should be distinguished in one respect from the other examples mentioned, which in any case form quite a mixed bag. Namely, each of the Fermat, fourcolor, and Kepler’s packing problems posed a standout challenge following extended efforts to settle them; meeting the challenge in each case required new ideas or approaches and intense work, obviously of different degrees. By contrast, Gödel’s theorems were simply unexpected, and their proofs, though requiring novel techniques, were not difficult on the scale of things. SetSolomon Feferman is professor of mathematics and philosophy, emeritus, at Stanford University. His email address is
BERNAYS AND SET THEORY
"... We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. ..."
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We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles.
Published by: NEWPHILSOC PUBLISHING 5 FALCONAR HOUSE
"... Newcastle Philosophy Society is a Registered Charity for the promotion of Philosophy (Registration Number 1106082). This is a critical / scholarly / creative work intended solely for educational and academic uses. Edition 1 first published November 2010. British Library CataloguinginPublication Da ..."
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Newcastle Philosophy Society is a Registered Charity for the promotion of Philosophy (Registration Number 1106082). This is a critical / scholarly / creative work intended solely for educational and academic uses. Edition 1 first published November 2010. British Library CataloguinginPublication Data A catalogue entry for this book is available from the British Library.
www.elsevier.com/locate/yaama Is complexity a source of incompleteness?
, 2004
"... In this paper we prove Chaitin’s “heuristic principle, ” the theorems of a finitelyspecified theory cannot be significantly more complex than the theory itself, for an appropriate measure of complexity. We show that the measure is invariant under the change of the Gödel numbering. For this measure, ..."
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In this paper we prove Chaitin’s “heuristic principle, ” the theorems of a finitelyspecified theory cannot be significantly more complex than the theory itself, for an appropriate measure of complexity. We show that the measure is invariant under the change of the Gödel numbering. For this measure, the theorems of a finitelyspecified, sound, consistent theory strong enough to formalize arithmetic which is arithmetically sound (like Zermelo–Fraenkel set theory with choice or Peano Arithmetic) have bounded complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. Previous results showing that incompleteness is not accidental, but ubiquitous are here reinforced in probabilistic terms: the probability that a true sentence of length n is provable in the theory tends to zero when n tends to infinity, while the probability that a sentence of length n is true is strictly positive. © 2004 Elsevier Inc. All rights reserved. 1.
Incompleteness: The Proof and Paradox of Kurt
, 2005
"... Popular books on mathematics play an important role in the lay public’s education. But as is known to anyone who has given a popular mathematics lecture or written about a famous theorem for an audience of nonmathematicians, doing justice to the mathematics in question is almost impossible in those ..."
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Popular books on mathematics play an important role in the lay public’s education. But as is known to anyone who has given a popular mathematics lecture or written about a famous theorem for an audience of nonmathematicians, doing justice to the mathematics in question is almost impossible in those circumstances. Rebecca Goldstein, the MacArthur Foundation fellow and author of The MindBody Problem (a novel which seems to be quite popular among mathematicians) attempts an even more difficult task in her short new book
The Liar Paradox
"... This essay delivers a serious message on life, at the core of reasoning, to wit, “I am a liar ” selfdestructive infests reasoning to undermine it to fill life with oxymora, then proposes a singing way out in music. This essay is naturally divided into three sections: one, how selfdestructive the l ..."
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This essay delivers a serious message on life, at the core of reasoning, to wit, “I am a liar ” selfdestructive infests reasoning to undermine it to fill life with oxymora, then proposes a singing way out in music. This essay is naturally divided into three sections: one, how selfdestructive the liar paradox is lifeubiquitous; two, the liar paradox as basic to logic and philosophy; and three, how to deal with the liar paradox in snakecharming, in “Tying. ” In short, these pages say that “I am a liar” selfdenies to selfdestroy, to demolish the structure of living (Section 1) and reasoning (Section 2). Such selfdemolition must be snakecharmed into deeply singing life and reasoning (Section 3).
The Gospel of Selfing: A Phenomenology of Sleep
"... Sleep is consciousness naturally folded back to itself in the selfcomehometoself, to find life nourished, renovated, and vitalized, all beyond objective management. Sleep can never be understood with direct conscious approach, but must be approached indirectly, implicatively, and alive coherent ..."
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Sleep is consciousness naturally folded back to itself in the selfcomehometoself, to find life nourished, renovated, and vitalized, all beyond objective management. Sleep can never be understood with direct conscious approach, but must be approached indirectly, implicatively, and alive coherently, as tried here. Sleep (A) is Spontaneity, (B) SelfFullness, and so (C) sleep is life’s Gospel of Selfing.