论文标题
一个学习问题，其一致性等效于不存在实用值可测量的红衣主教
A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals
论文作者
论文摘要
我们表明，$ k $ - 最近的邻居学习规则在公制空间$ x $中普遍一致，并且只有在每个可分离的$ x $的可分离子空间都普遍一致，并且$ x $的密度比每个真实的红衣主教都小。尤其是，在每个度量空间中，$ k $ -nn分类器在每个度量空间中都普遍一致，其可分离子空间在nagata和nagata的意义上是sigma-finite尺寸，并且仅当没有实现的可测量可测量的红衣主教。后一个假设与ZFC相对一致，但是在ZFC中不能证明这种基本主教的存在。我们的结果的灵感来自于2006年Cérou和Guyader在直观的严格水平上勾勒出的一个例子。
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.
