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Is Independence an Exception?
, 1994
"... Gödel's Incompleteness Theorem asserts that any sufficiently rich, sound, and recursively axiomatizable theory is incomplete. We show that, in a quite general topological sense, incompleteness is a rather common phenomenon: With respect to any reasonable topology the set of true and unprovable ..."
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Cited by 18 (12 self)
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Gödel's Incompleteness Theorem asserts that any sufficiently rich, sound, and recursively axiomatizable theory is incomplete. We show that, in a quite general topological sense, incompleteness is a rather common phenomenon: With respect to any reasonable topology the set of true and unprovable statements of such a theory is dense and in many cases even corare.
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.
A Quantum Mechanical Look At Time Travel and Free Will
, 2001
"... Consequences of the basic and most evident consistency requirementthat measured events cannot happen and not happen at the same timeare reviewed. Particular emphasis is given to event forecast and event control. As a consequence, particular, very general bounds on the forecast and control of e ..."
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Cited by 2 (2 self)
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Consequences of the basic and most evident consistency requirementthat measured events cannot happen and not happen at the same timeare reviewed. Particular emphasis is given to event forecast and event control. As a consequence, particular, very general bounds on the forecast and control of events within the known laws of physics result. These bounds are of a global, statistical nature and need not aect singular events or groups of events. We also present a quantum mechanical model of time travel and discuss chronology protection schemes. Such models impose restrictions upon certain capacities of event control.
Lisp ProgramSize Complexity II
, 1992
"... We present the informationtheoretic incompleteness theorems that arise in a theory of programsize complexity based on something close to real LISP. The complexity of a formal axiomatic system is defined to be the minimum size in characters of a LISP definition of the proofchecking function associa ..."
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Cited by 1 (1 self)
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We present the informationtheoretic incompleteness theorems that arise in a theory of programsize complexity based on something close to real LISP. The complexity of a formal axiomatic system is defined to be the minimum size in characters of a LISP definition of the proofchecking function associated with the formal system. Using this concrete and easy to understand definition, we show (a) that it is difficult to exhibit complex Sexpressions, and (b) that it is difficult to determine the bits of the LISP halting probability\Omega LISP . We also construct improved versions\Omega 0 LISP and\Omega 00 LISP of the LISP halting probability that asymptotically have maximum possible LISP complexity. Copyright c fl 1992, Elsevier Science Publishing Co., Inc., reprinted by permission. 2 G. J. Chaitin 1. Introduction The main incompleteness theorems of myAlgorithmic Information Theory monograph [1] are reformulated and proved here using a concrete and easytounderstand definition ...
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.