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Switch to: Citations
References in:
Incompleteness, complexity, randomness and beyond
Minds and Machines 12 (4):503-517 (2002)
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John Barrow is increasingly recognized as one of our most elegant and accomplished science writers, a brilliant commentator on cosmology, mathematics, and modern physics. Barrow now tackles the heady topic of impossibility, in perhaps his strongest book yet. Writing with grace and insight, Barrow argues convincingly that there are limits to human discovery, that there are things that are ultimately unknowable, undoable, or unreachable. He first examines the limits on scientific inquiry imposed by the deficiencies of the human mind: our (...) |
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Hao Wang was one of the few confidants of the great mathematician and logician Kurt Gödel. _A Logical Journey_ is a continuation of Wang's _Reflections on Gödel_ and also elaborates on discussions contained in _From Mathematics to Philosophy_. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem (...) |
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We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on Godel's first incompleteness theorem. |
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In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he (...) |
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The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of (...) |
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The papers gathered in this book were published over a period of more than twenty years in widely scattered journals. They led to the discovery of randomness in arithmetic which was presented in the recently published monograph on?Algorithmic Information Theory? by the author. There the strongest possible version of Gdel's incompleteness theorem, using an information-theoretic approach based on the size of computer programs, was discussed. The present book is intended as a companion volume to the monograph and it will serve (...) |
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This book presents one of the most intellectually challenging aspects of computer related mathematics/logic in a way which should make it accessible to a wider audience. The authors look at different types of reduction to show undecidability, but do so using the novel approach of conversation between three famous mathematicians - sometimes using their own words and sometimes in an adapted form. The authors are of international repute and they provide a modern and authoritative treatment of undecidability with special emphasis (...) |
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Recent findings in the computer sciences, discrete mathematics, formal logics and metamathematics have opened up a royal road for the investigation of undecidability and randomness in physics. A translation of these formal concepts yields a fresh look into diverse features of physical modelling such as quantum complementarity and the measurement problem, but also stipulates questions related to the necessity of the assumption of continua.Conversely, any computer may be perceived as a physical system: not only in the immediate sense of the (...) |
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many symbols. We define o, as the probability that an arbitrary machine be circular and we prove that o, is a random number that goes beyond.. No categories |
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In a famous lecture in 1900, David Hilbert listed 23 difficult problems he felt deserved the attention of mathematicians in the coming century. His conviction of the solvability of every mathematical problem was a powerful incentive to future generations: ``Wir müssen wissen. Wir werden wissen.'' (We must know. We will know.) Some of these problems were solved quickly, others might never be completed, but all have influenced mathematics. Later, Hilbert highlighted the need to clarify the methods of mathematical reasoning, using (...) |
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