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可数选择公理.
用法
作为应用
的例子,下面是所有无限集合是戴德金无限的一个证明(在
中):
- 设
是无限的。对于每个自然数
,设
是
的所有
元素子集的集合。因为
是无限的,每个
是非空的。对序列
应用
,便得到了序列(
),这里的每个
是有
个元素的
的子集。
- 集合
可能是相交的,但是我们可以定义
![{\displaystyle C_{0}=B_{0))](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc5725deeb6df9772dc5ec79710d186f42e7824)
是
与所有
的并集的差集,
。
- 明显的每个集合
都有至少1个和至多
个元素,而集合
是两两不相交的。再对序列
应用
,便得到了序列
,其中
。
- 所以所有
都是相异的,而
包含一个可数集合。定义把每个
映射到
的函数
(并固定所有
的其他元素),f是从
到
的一一映射,它不是满射,这证明了
是戴德金无限的。
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