Pitowsky's Kolmogorovian models and Super-Determinism

TL;DR

This paper analyzes Pitowsky's models that attempt to reconcile local hidden variables with quantum mechanics, arguing they are equivalent to super-determinism and physically irrelevant due to their reliance on nonstandard probability notions.

Contribution

It clarifies that Pitowsky's Kolmogorovian models are essentially super-deterministic, challenging their physical relevance and implications for set theory and quantum foundations.

Findings

Pitowsky's models employ nonstandard probability concepts.

These models are equivalent to super-determinism.

Physically relevant hidden variables require standard probability measures.

Abstract

In an attempt to demonstrate that local hidden variables are mathematically possible, Pitowsky constructed "spin-$\frac12$ functions" and later "Kolmogorovian models", which employs a nonstandard notion of probability. We describe Pitowsky's analysis and argue (with the benefit of hindsight) that his notion of hidden variables is in fact just super-determinism (and accordingly physically not relevant). Pitowsky's first construction uses the Continuum Hypothesis. Farah and Magidor took this as an indication that at some stage physics might give arguments for or against adopting specific new axioms of set theory. We would rather argue that it supports the opposing view, i.e., the widespread intuition "if you need a non-measurable function, it is physically irrelevant".