# 依賴選擇公理

## 應用

${\displaystyle {\mathsf {DC}}}$這條公理是${\displaystyle {\mathsf {AC}}}$的片斷，而在「必須於每一步都做出選擇」且「一些選擇無法在不仰賴先前選擇的情形下獨立做出」的狀況下證明「存在有可以可數長度的超限遞歸建構的序」列時，這條公理是必須的。

## 等價陳述

${\displaystyle {\mathsf {ZF}}}$的框架下，這公理也等價於勒文海姆–斯科倫定理[b][2]

${\displaystyle {\mathsf {DC}}}$${\displaystyle {\mathsf {ZF}}}$的框架下也與「所有有${\displaystyle \omega }$層且剪枝過的樹英语pruned tree都有」這陳述等價。

## 註解

1. ^ "The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." Bernays, Paul. Part III. Infinity and enumerability. Analysis. (PDF). Journal of Symbolic Logic. A system of axiomatic set theory. 1942, 7 (2): 65–89 [2022-07-23]. JSTOR 2266303. MR 0006333. doi:10.2307/2266303. （原始内容存档 (PDF)于2022-07-23）. The axiom of dependent choice is stated on p. 86.
2. ^ Moore states that "Principle of Dependent Choices ${\displaystyle \Rightarrow }$ Löwenheim–Skolem theorem" — that is, ${\displaystyle {\mathsf {DC}}}$ implies the Löwenheim–Skolem theorem. See table Moore, Gregory H. Zermelo's Axiom of Choice: Its origins, development, and influence. Springer. 1982: 325. ISBN 0-387-90670-3.

## 參考資料

1. ^ 「貝爾綱定理蘊含依賴選擇公理」─Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 1977, 25 (10): 933–934.
2. ^ The converse is proved in Boolos, George S.; Jeffrey, Richard C. Computability and Logic 3rd. Cambridge University Press. 1989: 155–156. ISBN 0-521-38026-X.
3. ^ Wolk, Elliot S., On the principle of dependent choices and some forms of Zorn's lemma 26 (3), Canadian Mathematical Bulletin: 365–367, 1983 [2022-07-23], , （原始内容存档于2022-07-23）
4. ^ 伯奈斯證明說依賴選擇公理蘊含可數選擇公理，相關資料可見於Bernays, Paul. Part III. Infinity and enumerability. Analysis. (PDF). Journal of Symbolic Logic. A system of axiomatic set theory. 1942, 7 (2): 65–89 [2022-07-23]. JSTOR 2266303. MR 0006333. doi:10.2307/2266303. （原始内容存档 (PDF)于2022-07-23）.的第86頁
5. ^ 對於可數選擇公理不蘊含依賴選擇公理這點，可見Jech, Thomas, The Axiom of Choice, North Holland: 130–131, 1973, ISBN 978-0-486-46624-8