标题： 可数集的大小 显示英文标题

标题： Sizes of Countable Sets

摘要： 本文引入了计数集大小的概念，该概念保留了部分-整体原则，并推广了有限集基数的概念。 自然数、整数、有理数及其所有子集、并集和笛卡尔积的大小都可以算法上枚举到一个元素，作为自然数序列。 该方法类似于Benci和Di Nasso（2019年）的Numerosity理论，但相比之下，它受到Bolzano在《无穷悖论》中关于无穷级数概念的启发，它是构造性的因为它不使用超滤器，并且集合大小是唯一确定的。 结果大多与Numerosity理论的结果一致，但有些不同，例如有理数的大小。 然而，集合大小只是部分而非线性排序的。 \emph{有来有往。} 显示英文摘要

摘要： The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their subsets, unions, and Cartesian products are algorithmically enumerable up to one element as sequences of natural numbers. The method is similar to that of Theory of Numerosities of Benci and Di Nasso 2019) but in comparison, it is motivated by Bolzano's concept of infinite series from his Paradoxes of the Infinite, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree with those of Theory of Numerosities, but some differ, such as the size of rational numbers. However, set sizes are just partially and not linearly ordered. \emph{Quid pro quo.}