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 C\'erou and Guyader in 2006 at an intuitive level of rigour.

We show that a recently proposed 1-nearest-neighbor-based multiclass learning algorithm is universally strongly Bayes consistent in all metric spaces where such Bayes consistency is possible, making it an optimistically universal Bayes-consistent learner. This is the first learning algorithm known to enjoy this property; by comparison, $k$-NN and its variants are not generally universally Bayes consistent, except under additional structural assumptions, such as an inner product, a norm, finite doubling dimension, or a Besicovitch-type property. The metric spaces in which universal Bayes consistency is possible are the essentially separable ones --- a new notion that we define, which is more general than standard separability. The existence of metric spaces that are not essentially separable is independent of the ZFC axioms of set theory. We prove that essential separability exactly characterizes the existence of a universal Bayes-consistent learner for the given metric space. In particular, this yields the first impossibility result for universal Bayes consistency. Taken together, these positive and negative results resolve the open problems posed in Kontorovich, Sabato, Weiss (2017).