# 基数 (数学)

## 维基百科，自由的百科全书

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

## 动机

${\displaystyle 1\to a}$
${\displaystyle 2\to b}$
${\displaystyle 3\to c}$

${\displaystyle 1\longleftrightarrow 2}$
${\displaystyle 2\longleftrightarrow 3}$
${\displaystyle 3\longleftrightarrow 4}$
...
${\displaystyle n\longleftrightarrow n+1}$
...

## 定义

（若不接受选择公理，我们也可对非良序集${\displaystyle X}$定义基数，就是所有与${\displaystyle X}$等势的集的阶中最小者。）

## 基数算术

${\displaystyle |X|+|Y|=|X+Y|}$

${\displaystyle X}$${\displaystyle Y}$不相交，则 ${\displaystyle |X|+|Y|=|X\cup Y|}$

${\displaystyle |X||Y|=|X\times Y|}$

${\displaystyle |X|^{|Y|}=|X^{Y}|}$

• 加法和乘法是可交换的，即 ${\displaystyle |X|+|Y|=|Y|+|X|}$${\displaystyle |X||Y|=|Y||X|}$
• 加法和乘法符合结合律${\displaystyle (|X|+|Y|)+|Z|=|X|+(|Y|+|Z|)}$${\displaystyle (|X||Y|)|Z|=|X|(|Y||Z|)}$
• 分配律，即 ${\displaystyle (|X|+|Y|)|Z|=|X||Z|+|Y||Z|}$
• ${\displaystyle |X|^{|Y|+|Z|}=|X|^{|Y|}|X|^{|Z|))$
• ${\displaystyle |X|^{|Y||Z|}=(|X|^{|Y|})^{|Z|))$
• ${\displaystyle (|X||Y|)^{|Z|}=|X|^{|Z|}|Y|^{|Z|))$

${\displaystyle |X|+|Y|=|X||Y|=\max\{|X|,|Y|\))$

• ${\displaystyle |X|^{0}=1}$（特别地，${\displaystyle 0^{0}=1}$）。
• ${\displaystyle 0^{|Y|}=0}$，若${\displaystyle Y}$非空。
• ${\displaystyle 1^{|Y|}=1}$
• ${\displaystyle |X|\leq |Y|}$，则 ${\displaystyle |X|^{|Z|}\leq |Y|^{|Z|))$
• ${\displaystyle |X|}$${\displaystyle |Y|}$ 俱有限且大于1，而${\displaystyle Z}$是无穷集，则 ${\displaystyle |X|^{|Z|}=|Y|^{|Z|))$
• 若X是无穷而${\displaystyle Y}$是有限及非空，则 ${\displaystyle |X|^{|Y|}=|X|}$

## 参考文献

• Hahn, Hans, Infinity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. New York: Simon and Schuster, 1956.
• Halmos, Paul, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

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