公理集合论？

公理集合论是19世纪末康托尔创立的。由于罗素悖论及一系列别的悖论，使得原始的朴素的集合理论被迫采用数学中最常用的办法——公理化方法——来避免悖论。我们来看看维基百科上怎么描述集合公理：
1. Axiom of extensionality: Two sets are the same if and only if they have the same elements.
2. Axiom of empty set: There is a set with no elements. We will use {} to denote this empty set.
3. Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
4. Axiom of union: Every set has a union. That is, for any set x there is a set y whose elements are precisely the elements of the elements of x.
5. Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.
6. Axiom of separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds.
7. Axiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements.
8. Axiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.
9. Axiom of regularity (or axiom of foundation): Every non-empty set x contains some element y such that x and y are disjoint sets.
10. Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.
看起来都比较简单，但是却没有办法证明完备性。同时，无论第十条公理（即选择公理）采用哪一种方案，都会推导出与现实矛盾的结论出来。但是，每一条公理看起来都是符合直觉的。难怪康托尔最后会疯掉。现代数学中也不全都会采用Zermelo的选择公理，现在存在的版本还有Von Neumann-Bernays-Gödel 集合理论 (NBG), Kripke-Platek 集合理论 (KP), Kripke-Platek 结合理论连同 urelements (KPU) , Morse-Kelley 集合理论. ZFC（也就是上面提到的十公理）公理系统的独立性。 //。。。有了集合公理体系，其他所有的数学概念都可以被他们构造出来，比如：数字、离散和连续、序、关系和函数等。举例来说，序关系就可以通过集合中的序对这个概念构造出来。作为聪明的中国人，我就不需要一步一步地证明了吧。提示一下：这儿可以用递归，如果对于无穷有序集合，可以使用无穷递归。另一个例子就是自然数了，自然数可以通过集合加上后继这个概念构造出来，这一次我要给出步骤了： 0 = {} 1 = { 0 } = { {} } 2 = { 0, 1 } = { {}, { {} } } 3 = { 0, 1, 2 } = { {}, { {} }, { {}, { {} } } } ... 唔，看起来还不错：），呵呵。像关系或者更特殊的关系——函数，我们不妨看看它们的定义，是不是通过集合构造出来的。当然，一旦我们深入研究更抽象的东西，比如范畴，或者范畴之范畴，我们就需要更精细的集合公理了，但是不管怎么说，公理集合论是现代数学的基石确是无可否认的。