Incompleteness Theorems for Random Reals (1987)
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@MISC{Chaitin87incompletenesstheorems,
author = {G. J. Chaitin},
title = {Incompleteness Theorems for Random Reals},
year = {1987}
}
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Abstract
We obtain some dramatic results using statistical mechanics--thermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property that if one varies a parameter and asks whether the number of solutions is finite or infinite, the answer to this question is indistinguishable from the result of independent tosses of a fair coin. This yields a number of powerful Godel incompleteness-type results concerning the limitations of the axiomatic method, in which entropy--information measures are used. c fl 1987 Academic Press, Inc. 2 G. J. Chaitin 1. Introduction It is now half a century since Turing published his remarkable paper On Computable Numbers, with an Application to the Entscheidungsproblem (Turing [15]). In that paper Turing constructs a universal Turing machine that can simulate any other Turing machine. He also use...
