A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals
Vladimir G. Pestov
TL;DR
The paper addresses whether the $k$-NN classifier is universally consistent in non-separable metric spaces and shows this property is tightly linked to set-theoretic foundations, specifically the existence of real-valued measurable cardinals. The main result states that universal consistency in a space Omega holds precisely when it holds in every separable subspace and the density $d(Omega)$ is smaller than the smallest real-valued measurable cardinal. The analysis blends large-cardinal set theory (Ramsey properties for two-valued and atomlessly measurable cardinals, Gitik–Shelah results) with metric-dimension theory (Nagata dimension, sigma-finite dimensionality) and leverages existing separable-space results to obtain a complete characterization. As a consequence, the work yields concrete corollaries for exotic spaces such as metric hedgehogs and spaces like $c_{00}(\Gamma)$, clarifying when universal consistency can occur in highly non-separable domains and highlighting the independence of certain learning questions from ZFC assumptions.
Abstract
We show that the $k$-nearest neighbour learning rule is universally consistent in a metric space $X$ if and only if it is universally consistent in every separable subspace of $X$ and the density of $X$ is less than every real-measurable cardinal. In particular, the $k$-NN classifier is universally consistent in every metric space whose separable subspaces are sigma-finite dimensional in the sense of Nagata and Preiss if and only if there are no real-valued measurable cardinals. The latter assumption is relatively consistent with ZFC, however the consistency of the existence of such cardinals cannot be proved within ZFC. Our results were inspired by an example sketched by Cérou and Guyader in 2006 at an intuitive level of rigour.