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Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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Proofs Without Syntax
 Annals of Mathematics
"... [M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatori ..."
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[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graphtheoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.
What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end
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
ITS HISTORY IN THE WORK AND WRITINGS OF
"... Jean van Heijenoort (hereafter JvH) is mainly known among logicians as the editor of From Frege to Gödel (hereafter FtG). Most logicians have also heard about his peculiar life which has contributed to turning him into a legendary personage. This aspect became well ..."
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Jean van Heijenoort (hereafter JvH) is mainly known among logicians as the editor of From Frege to Gödel (hereafter FtG). Most logicians have also heard about his peculiar life which has contributed to turning him into a legendary personage. This aspect became well
ARTICLE NO. DR980467 Hierarchical Complexity of Tasks Shows the Existence of Developmental Stages Michael Lamport Commons
"... The major purpose of this paper is to introduce the notion of the order of hierarchical complexity of tasks. Order of hierarchical complexity is a way of conceptualizing information in terms of the power required to complete a task or solve a problem. It is orthogonal to the notion of information c ..."
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The major purpose of this paper is to introduce the notion of the order of hierarchical complexity of tasks. Order of hierarchical complexity is a way of conceptualizing information in terms of the power required to complete a task or solve a problem. It is orthogonal to the notion of information coded as bits in traditional information theory. Because every task (whether experimental or everyday) that individuals engage in has an order of hierarchical complexity associated with it, this notion of hierarchical complexity has broad implications both within developmental psychology and beyond it in such fields as information science. Within developmental psychology, traditional stage theory has been criticized for not showing that stages exist as anything more than ad hoc descriptions of sequential changes in human behavior
THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING
"... www.elsevier.com/locate/jlap Combining programs and state machines ..."