  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.
