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From Heisenberg to Gödel via Chaitin
, 2008
"... In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa”. Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. A ..."
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Cited by 11 (9 self)
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In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa”. Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle ” which implies Chaitin’s informationtheoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. This fact supports the conjecture that uncertainty implies randomness not only in mathematics, but also in physics.
Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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Centre for Discrete Mathematics and Theoretical Computer ScienceSearching for Spanning kCaterpillars and kTrees
, 2008
"... We consider the problems of finding spanning kcaterpillars and ktrees in graphs. We first show that the problem of whether a graph has a spanning kcaterpillar is NPcomplete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1caterpillar in a graph with treewidth k. Also ..."
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We consider the problems of finding spanning kcaterpillars and ktrees in graphs. We first show that the problem of whether a graph has a spanning kcaterpillar is NPcomplete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1caterpillar in a graph with treewidth k. Also, as a generalized versions of the depthfirst search and the breadthfirst search algorithms, we introduce the ktree search (KTS) algorithm and we use it in a heuristic algorithm for finding a large kcaterpillar in a graph. 1
Dyson Statements that Are Likely to Be True but Unprovable
, 2008
"... Gödel’s Incompleteness Theorem states that every finitelyspecified, sound, theory which is strong enough to include arithmetic cannot be both consistent and complete. In particular, if the theory is consistent, then it is incomplete, i.e. it contains statements that cannot be proved nor disproved. ..."
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Gödel’s Incompleteness Theorem states that every finitelyspecified, sound, theory which is strong enough to include arithmetic cannot be both consistent and complete. In particular, if the theory is consistent, then it is incomplete, i.e. it contains statements that cannot be proved nor disproved. If we adopt a semantic criterion (a definition of true statements), then the theory contains true and unprovable statements. The set of is true and unprovable statements is large in both topological sense [?] and probabilistic sense [?] (see more in [?]). Are there examples of true and unprovable statements? Yes, but they are not simple [?]. In [?], p. 86, Dyson stated: