The structure of the real line in quantum mechanics and cosmology

Abstract

We discuss the recently proposed model, where the spacetime in large scales is parametrized by the usual real line R, while at small (quantum mechanical) scales, the space is parametrized by the real numbers RM from some formal model M of Zermelo–Fraenkel set theory. We argue that the set-theoretic forcing is an important ingredient of the shift from micro- to macroscale. The set RM, describing the space at the Planck era, is merely a meager subset of R. It is Lebesgue non-measurable and all its measurable subsets have Lebesgue measure 0. According to this, the contributions to the cosmological constant from the zero-point energies of quantum fields vanish. Moreover, the emerged irregularities in the real line can be considered as the source of the primordial quantum fluctuations.