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Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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Propagation of partial randomness
"... Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongl ..."
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Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongly frandom relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including nonKtriviality and autocomplexity. We prove that frandomness relative to a PAdegree implies strong frandomness, but frandomness does not imply frandomness relative to a PAdegree. Keywords: partial randomness, effective Hausdorff dimension, MartinLöf randomness, Kolmogorov complexity, models of arithmetic.
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.
Theoretical Computer Science 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, t ..."
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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 ZermeloFraenkel 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. 1
Incompleteness, Complexity, Randomness and Beyond
, 2001
"... The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified i ..."
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The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified in a cryptographical or allegorical manner; such a justification is verbal and, ex hypothesi, already figures in the Library.... The certitude that some shelf in some hexagon held precious books and that these precious books were inaccessible seemed almost intolerable. A blasphemous sect suggested that... all men should juggle letters and symbols until they constructed, by an improbable gift of chance, these canonical books... but the Library is... useless, incorruptible, secret. Jorge Luis Borges, “The Library of Babel” Gödel’s Incompleteness Theorems have the same scientific status as Einstein’s principle of relativity, Heisenberg’s uncertainty principle, and Watson and Crick’s double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an informationtheoretic approach to randomness and recent developments in quantum computing.
Explicit Finitism
, 2001
"... this paper we show how the problem can be stated in a more interesting form, as being about sequences J i of subsets of the real numbers tending to a limit J R, and pose some open questions about J ..."
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this paper we show how the problem can be stated in a more interesting form, as being about sequences J i of subsets of the real numbers tending to a limit J R, and pose some open questions about J
Panu Raatikainen THE PROBLEM OF THE SIMPLEST DIOPHANTINE REPRESENTATION
"... Gregory Chaitin’s informationtheoretic incompleteness result (Chaitin 1974a,b) has received extraordinary attention; it is apparently one of the most widely known recent logical results. 1 Roughly, it says that for every formalized theory there is a finite constant c such that ..."
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Gregory Chaitin’s informationtheoretic incompleteness result (Chaitin 1974a,b) has received extraordinary attention; it is apparently one of the most widely known recent logical results. 1 Roughly, it says that for every formalized theory there is a finite constant c such that
PANU RAATIKAINEN ALGORITHMIC INFORMATION THEORY AND UNDECIDABILITY
"... Algorithmic information theory, or the theory of Kolmogorov complexity, has become an extraordinarily popular theory, and this is no doubt due, in some part, to the fame of Chaitin’s incompleteness results arising from this field. Actually, there are two rather different results by Chaitin: the ..."
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Algorithmic information theory, or the theory of Kolmogorov complexity, has become an extraordinarily popular theory, and this is no doubt due, in some part, to the fame of Chaitin’s incompleteness results arising from this field. Actually, there are two rather different results by Chaitin: the
The Scope of Gödel’s First Incompleteness Theorem
"... Abstract. Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of ..."
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Abstract. Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of