# 选择公理

(Si) 是一个以实数集R指标集集族；也就是说，对每一个实数i，均存在一个集合 Si，如图所示。每一个集合包含至少一个（可能是无限个）元素。选择公理可以断言，我们可以从每一个集合中选择一个元素，组成一个在R上的索引族(xi)，这里xi∈SiiR。一般情况下，指标集可以是任意集合I，而不仅仅是R

## 陈述

${\displaystyle \forall X\left[\emptyset \notin X\implies \exists f\colon X\rightarrow \bigcup X\quad \forall A\in X\,(f(A)\in A)\right]\,.}$

${\displaystyle X}$ 是一个集族，则存在着在${\displaystyle X}$ 上定义的一个选择函数${\displaystyle f}$

## 参考文献

### 引用

1. ^
2. ^ Jech, 1977, p. 348ff; Martin-Löf 2008, p. 210. According to Mendelson 1964，第201页: The status of the Axiom of Choice has become less controversial in recent years. To most mathematicians it seems quite plausible and it has so many important applications in practically all branches of mathematics that not to accept it would seem to be a wilful hobbling of the practicing mathematician.
3. ^ Patrick Suppes, "Axiomatic Set Theory", Dover, 1972 (1960), ISBN 978-0-486-61630-8, pp 240
4. ^ Per Martin-Löf, Intuitionistic type theory, 1980. Anne Sjerp Troelstra, Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, 1973.
5. ^ Errett Bishop and Douglas S. Bridges, Constructive analysis, Springer-Verlag, 1985.
6. ^ Per Martin-Löf, "100 Years of Zermelo’s Axiom of Choice: What was the Problem with It?", The Computer Journal (2006) 49 (3): 345-350. doi: 10.1093/comjnl/bxh162
7. ^ Fred Richman, “Constructive mathematics without choice”, in: Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum (P. Schuster et al., eds), Synthèse Library 306, 199–205, Kluwer Academic Publishers, Amsterdam, 2001.
8. ^ Devlin, Keith. Constructibility. Springer-Verlag. 1984. ISBN 3-540-13258-9.

### 来源

Translated in: Jean van Heijenoort, 2002. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. New edition. Harvard Univ. Press. ISBN 978-0-674-32449-7.
• 1904. "Proof that every set can be well-ordered," 139-41.
• 1908. "Investigations in the foundations of set theory I," 199-215.
• Gregory H Moore, "Zermelo's axiom of choice, Its origins, development and influence", Springer; 1982. ISBN 978-0-387-90670-6.
• Paul Howard and Jean Rubin, "Consequences of the Axiom of Choice". Mathematical Surveys and Monographs 59; American Mathematical Society; 1998.

## 外部链接

• Paul Howard at EMU有很多人仍然在为选择公理和它的推论而乐此不疲地工作。如果你有兴趣了解更多内容，请参考这个网站。