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

TL;DR

This paper demonstrates an inherent uncertainty in physical theories related to the failure of the physical Church-Turing thesis, showing limitations in decision procedures for non-recursive real outputs and implications for falsifiability.

Contribution

It introduces a decision process that can handle non-recursive real outputs, challenging assumptions about the computability of physical theories.

Findings

No computable decision procedure can reliably identify non-recursive real outputs.

The decision process can determine if the mean of an i.i.d. sequence belongs to a specific Δ₂ set.

Implications for the falsifiability of physical theories and the limits of computational models in physics.

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.