Indeterminism and Undecidability

TL;DR

This paper argues that quantum indeterminism can be demonstrated through Chaitin's incompleteness theorem, showing that quantum measurement outcomes are almost surely 1-random, thus stronger than uncomputability, challenging deterministic hidden variable theories.

Contribution

It introduces a novel proof of quantum indeterminism based on algorithmic randomness and incompleteness, extending the empirical content beyond Bell's theorem.

Findings

Quantum outcomes are 1-random sequences, almost surely.

Bell's theorem leaves loopholes by only considering relative frequencies.

Quantum randomness exceeds mere uncomputability, implying indeterminism.

Abstract

The aim of this paper is to argue that the (alleged) indeterminism of quantum mechanics, claimed by adherents of the Copenhagen interpretation since Born (1926), can be proved from Chaitin's follow-up to Goedel's (first) incompleteness theorem. In comparison, Bell's (1964) theorem as well as the so-called free will theorem-originally due to Heywood and Redhead (1983)-left two loopholes for deterministic hidden variable theories, namely giving up either locality (more precisely: local contextuality, as in Bohmian mechanics) or free choice (i.e. uncorrelated measurement settings, as in 't Hooft's cellular automaton interpretation of quantum mechanics). The main point is that Bell and others did not exploit the full empirical content of quantum mechanics, which consists of long series of outcomes of repeated measurements (idealized as infinite binary sequences): their arguments only used the long-run relative frequencies derived from such series, and hence merely asked hidden variable theories to reproduce single-case Born probabilities defined by certain entangled bipartite states. If we idealize binary outcome strings of a fair quantum coin flip as infinite sequences, quantum mechanics predicts that these typically (i.e.\ almost surely) have a property called 1-randomness in logic, which is much stronger than uncomputability. This is the key to my claim, which is admittedly based on a stronger (yet compelling) notion of determinism than what is common in the literature on hidden variable theories.