空集公理
Multi tool use
在集合论中，空集公理是 Zermelo-Fraenkel 集合论的公理之一。
形式陈述
在 Zermelo-Fraenkel 公理的形式语言中，这个公理读做：
- ∃A,∀x:¬(x∈A){displaystyle exists A,forall x:lnot (xin A)}
换句话说：
有着一个集合使得「没有集合」是它的元素。
解释
我们可以使用外延公理来证明只有一个这样的集合。因为它是唯一的，我们可以簡單名之為空集，并將其標記为 {} 或 ∅{displaystyle varnothing }
。因此这个公理的本质是：
- 存在一个空集。
空集公理一般被认为是无可争议的，它或它的等价命題出现在任何可替代的集合论的公理化中。
在 ZF 的某些陳述版本中，空集公理实际上在无穷公理中是重复的。换句话说，有不预設空集存在的另一种公理版本。还有，以一常量符号表示空集的話，藉此可以把其他 ZF 公理重寫成更簡潔的版本；那么无穷公理也會用到这个符号而不要求它是空的，尽管需要空集公理来表明它实际上是空的。
而且，在那些不包含无穷集合的集合论中，空集公理仍是需要的。就是说，使用分离公理模式，声称任何集合存在的任何公理都蕴涵空集公理。
引用
- Paul Halmos, 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).
- Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
- Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
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