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**1 - 8**of**8**### Is Complexity a Source of Incompleteness?

- IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004

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### The Gödel Paradox and Wittgenstein’s Reasons 1. The Implausibile Wittgenstein Wittgenstein’s notorious comments on Gödel’s First Incompleteness Theorem in the

"... Remarks on the Foundations of Mathematics were dismissed by early commentators, such as Kreisel, Anderson, Dummett, and Bernays, as an unfortunate episode in the career of a great philosopher. It appears that Wittgenstein had in his sights only the informal account of the Theorem, presented by Gödel ..."

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Remarks on the Foundations of Mathematics were dismissed by early commentators, such as Kreisel, Anderson, Dummett, and Bernays, as an unfortunate episode in the career of a great philosopher. It appears that Wittgenstein had in his sights only the informal account of the Theorem, presented by Gödel in the introduction of his celebrated 1931 paper, and was misguided by it (not that he was the only one: because of the misunderstandings it originated, Helmer said that that exposition “without any claim to complete precision”1 is the only mistake in Gödel’s paper). It is claimed that Wittgenstein erroneously considered essential the natural language interpretation of the Gödel sentence, whose undecidability within (the modified system considered by Gödel, taken from) Russell and Whitehead’s Principia mathematica is at the core of the First Theorem, as claiming “I am not provable”. On the contrary, Gödel’s proof can be framed in syntactic terms in which no extra-mathematical interpretation of the formulas is required. Commentators were particularly struck by the fact that Wittgenstein seems to take the Gödel formula as a paradoxical sentence, not too different from the usual Liar – and Gödel’s proof itself, therefore, as the deduction of an inconsistency: 11. Let us suppose I prove the unprovability (in Russell’s system) of P; then by this proof I have proved P. Now if this proof were one in Russell’s system – I should in this case have proved at once that it belonged and did not belong to Russell’s system. – That is what comes of making up such

### Arithmetic and the Incompleteness Theorems

, 2000

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### 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 finitely-specified 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 finitely-specified 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 finitely-specified, 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 finitely-specified, 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. 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 twenty-five 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 twenty-five 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 information-theoretic approach to randomness and recent developments in quantum computing.