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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.
Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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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
A coinductive approach to verified exact real number computation. 2009. To appear
 Proceedings of Automated Verification of Critical Systems (AVOCS), Gregynog
"... Abstract: We present an approach to verified programs for exact real number computation that is based on inductive and coinductive definitions and program extraction from proofs. We informally discuss the theoretical background of this method and give examples of extracted programs implementing th ..."
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Abstract: We present an approach to verified programs for exact real number computation that is based on inductive and coinductive definitions and program extraction from proofs. We informally discuss the theoretical background of this method and give examples of extracted programs implementing the translation between the representation by fast converging rational Cauchy sequences and the signed binary digit representations of real numbers.
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
Does the Category of Multisets Require a Larger Universe than that of the Category of Sets?
"... Copyright © 2013 Dasharath Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In section 1, the concept of a category is br ..."
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Copyright © 2013 Dasharath Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In section 1, the concept of a category is briefly described. In section 2, it is elaborated how the concept of category is naturally intertwined with the existence of a universe of discourse much larger than what is otherwise sufficient for a large part of mathematics.
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