Entropy, heat, and Gödel incompleteness

Abstract

Irreversible phenomena -such as the production of entropy and heat -arise from fundamental reversible dynamics because the forward dynamics is too complex, in the sense that it becomes impossible to provide the necessary information to keep track of the dynamics. On a heuristic level, this is well captured by coarse graining. We suggest that on a fundamental level the impossibility to provide the necessary information might be related to the incompleteness results of Gödel. This would hold interesting implications for both, mathematics and physics.

A Set-Theoretic Analysis of the Black Hole Entropy Puzzle

“…Then (8) can be interpreted as an argument for the "physical origin" (cf. Heisenberg uncertainity) of what we have called the inherent fuzziness or uncertainty within the set of real numbers [20,21]. This interpretation then could explain the expected independence of the set-theoretic entropy (4) of any axiomatic system S which is in accordance with the universal character of Theorem 2.1.…”

“…the set ℝ of real numbers or more generally any differentiable manifold, can be interpreted as a non-zero entropy of the arithmetical continuum (cf. [20,21]), quantitatively captured by (4) at least in the compact case.…”

“…hence does not carry essential information. A remarkable observation about Kolmogorov complexity is the following result: Theorem 2.1 (Chaitin's version of Gödel's first incompleteness theorem [4]) For any (sufficiently rich, consistent, recursively enumerable) axiomatic system S based on a first order language L there exists a natural number 0 < N S < +∞ such that there exists no real number x for which the proposition is provable within S. ◻ Motivated in various ways by [10,20,21] we interpret this quite surprising mathematical fact from our viewpoint as follows: taking into account that the only known property of a generic real number which fully characterizes it is its existing…”

On a Possibly Pure Set-Theoretic Contribution to Black Hole Entropy

“…A remarkable observation about Kolmogorov complexity is the following result [3]: Theorem 2.1. (Chaitin's version of Gödel's first incompleteness theorem) For any (sufficiently rich, consistent, recursively enumerable) axiomatic system S based on a first order language L there exists a natural number 0 < N S < +∞ such that there exists no real number x for which the proposition K(x) ≧ N S is provable within S. ✸ Motivated in various ways by [7,12,13] we interpret this quite surprising mathematical fact from our viewpoint as follows: Taking into account that the only known property of a generic real number which fully characterizes it is its existing (decimal) expansion, but the Kolmogorov complexity of this expansion hence the expansion itself generally is not fully determinable (by proving theorems on it in an axiomatic system), there is in general no way, using standard mathematical tools in the broadest sense, to "sharply pick" any element from the arithmetical continuum. Consequently, from the viewpoint of "effective mathematical activity", the structure of the arithmetical continuum i.e.…”

“…the set R of real numbers or gemetrically speaking the real line or more generally any differentiable manifold, can be interpreted as a non-zero entropy of the arithmetical continuum (cf. [12,13]).…”