An experimental uncertainty implied by failure of the physical Church-Turing thesis

Abstract

In this paper we prove that given a black box assumed to generate bits of a given non-recursive real $\Omega$ there is no computable decision procedure generating sequences of decisions such that if the output is indeed $\Omega$ the process eventually accepts the hypothesis while if the output is different than $\Omega$ than the procedure will eventually reject the hypothesis from a certain point on. Our decision concept does not require full certainty regarding the correctness of the decision at any point, thus better represents the validation process of physical theories. The theorem has strong implications on the falsifiability of physical theories entailing the failure of the physical Church Turing thesis. Finally we show that our decision process enables to decide whether the mean of an i.i.d. sequence of reals belongs to a specific $\Delta_2$ set of integers. This significantly strengthens the effective version of the Cover-Koplowitz theorem, beyond computable sequences of reals.