Quantum gravity from descriptive set theory

Abstract

We start from Hilbert’s criticism of the axioms of classical geometry and the possibility of abandoning the Archimedean axiom. Subsequently we proceed to the physical possibility of a fundamental limitation on the smallest length connected to certain singular points in spacetime and below which measurements become meaningless, Finally we arrive at the conclusion that maximising the Hawking–Bekenstein informational content of spacetime makes the existence of a transfinite geometry for physical “spacetime” not only plausible but probably inevitable. The main part of the paper is then concerned with a proposal for a mathematical description of a transfinite, non-Archimedean geometry using descriptive set theory. Nevertheless, and despite all abstract mathematics, we remain quite close to similar lines of investigation initiated by physicists like A. Wheeler, D. Finkelstein and G. ‘tHooft. In particular we introduce a logarithmic gauge transformation linking classical gravity with the electro weak via a version of informational entropy. That way we may claim to have accomplished an important step towards a general theory of quantum gravity using ε (∞) and complexity theory and finding that α ̄ G =(2) α ̄ ew −1 ≅(1.7)(10) 38 where α ̄ G is the dimensionless Newton gravity constant, and α ̄ ew ≃128 is the fine structure constant at the electro weak scale.