Logical independence and quantum randomness

TL;DR

This paper establishes a connection between logical independence and quantum randomness, demonstrating that quantum measurements can reveal logical dependencies and independencies of propositions within encoded axiomatic systems.

Contribution

It introduces a method to test logical independence experimentally using quantum systems and measurements, linking logical concepts with quantum physics.

Findings

Quantum states encode axioms and reveal logical dependencies.

Quantum measurements produce random outcomes for independent propositions.

The approach explains quantum randomness through logical independence without Gödel's incompleteness.

Abstract

We propose a link between logical independence and quantum physics. We demonstrate that quantum systems in the eigenstates of Pauli group operators are capable of encoding mathematical axioms and show that Pauli group quantum measurements are capable of revealing whether or not a given proposition is logically dependent on the axiomatic system. Whenever a mathematical proposition is logically independent of the axioms encoded in the measured state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of logical independence. Conversely, it also allows for an explanation of the probabilities of random outcomes observed in Pauli group measurements from logical independence without invoking quantum theory. The axiomatic systems we study can be completed and are therefore not subject to Goedel's incompleteness theorem.