如何比较两个无限集的元素“个数”？

How to Compare the “Number” of Elements of Two Infinite Sets?

摘要: 2019年人教版高中数学教科书必修第一册有一则“阅读与思考”材料–集合中元素的个数，其指出：有限集合中元素的个数可以一一数出来，便能通过比较自然数的大小直接比较有限集合元素个数的多少。而对于元素个数无限的集合，如N={0,1,2,…,n,…}，A={0,2,4,…,2n,…}无法一一数出集合中元素的个数，又该如何比较它们元素“个数”的多少呢？本文尝试类比比较两个有限集元素个数的方法，探讨如何比较两个无限集的元素“个数”。

Abstract: The first volume of the compulsory high school mathematics textbook of the 2019 People’s Education Edition has a “reading and thinking” material—the number of elements in the set, which points out that the number of elements in the finite set can be counted one by one, and the number of elements in the finite set can be directly compared by comparing the size of the natural number. For a set with an infinite number of elements, such as N={0,1,2,…,n,…} and A={0,2,4,…,2n,…}, the number of elements in the set can’t be counted one by one, so how to compare the number of their elements? This paper tries to compare the number of elements of two finite sets by analogy, and discusses how to compare the “number” of elements of two infinite sets.