We study families of functions and linear orders which separate countable subsets of the continuum from points. As an application, we show that the order dimension of the Turing degrees, denoted ${\mathfrak{dim}}_T$, cannot be decided in ZFC. We also provide a combinatorial description of ${\mathfrak{dim}}_T$ and show that the Turing degrees have the largest order dimension among all locally countable partial orders of size continuum. Finally, we prove that it is consistent that the number of linear orders needed to separate countable subsets of the continuum from points is strictly smaller than the number of functions necessary for separating them.

中文翻译：

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分离图灵度的族和阶维

我们研究了将连续体的可数子集与点分开的函数族和线性阶。作为一个应用程序，我们表明图灵度的阶数维表示为${\mathfrak{暗淡}}_{Ť}$，无法在ZFC中确定。我们还提供了以下内容的组合说明：${\mathfrak{暗淡}}_{Ť}$并表明在所有局部可数的大小连续体的局部数量级中，图灵度具有最大的数量级。最后，我们证明了一致的一点，即从点分离连续谱的可数子集所需的线性阶数严格小于将它们分离所需的函数数。