A set-theoretic analysis of the black hole entropy puzzle

Motivated by the known mathematical and physical problems arising from the current mathematical formalization of the physical spatio-temporal continuum, as a substantial technical clarification of our earlier attempt, the aim in this paper is twofold. Firstly, by interpreting Chaitin's variant of G\"odel's first incompleteness theorem as an inherent uncertainty or fuzziness present in the set of real numbers, a set-theoretic entropy is assigned to it using the Kullback--Leibler relative entropy of a pair of Riemannian manifolds. Then exploiting the non-negativity of this relative entropy an abstract Hawking-like area theorem is derived. Secondly, by analyzing Noether's theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a black hole is modeled by the set of real numbers in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole (proportional to the area of its ``instantaneous'' event horizon), indicating that this apparently physical quantity might have a pure set-theoretic origin, too.