怎么说呢,其实关于罗素悖论实际上已经被证明是不存在集合论里了,而不是把它变得公理化。以下是定理 Cantor's Theorem:If A is any set, then there is no surjection of A onto the set P(A) (power set of A) of all subset of A. Proof: Suppose contrary that f: A-> P(A) is a surjection. Since f(a) is a subset of A, either a belong to f(a) or it does not belong to this set. Let D = {a in A: a not in f(a)}. Since D is a subset of A, if f is a surjection, then D = f(a_0) for some a_0 in A. We must have either a_0 in D or a_0 not in D. If a_0 in D, then since D = f(a), we must have a_0 in f(a_0), contrary to the definition of D. Similarly, if a_0 not in D, then a_0 not in f(a_0) so that a_0 in D, which is also a contradiction. Therefore, f cannot be a surjection. 然后哥德尔不完备定理不是说的是哥德巴赫猜想,黎曼猜想不可证,而是说不可能用有限公理去证明所有的数学命题。毕竟人类能发现的公理都是有限的,无论数学发展得多高,总存在某些命题不可证。黎曼猜想,哥德巴赫猜想可能十几百年后因为有新的公理和工具所以得到证明,但是也会因为新公理的诞生导致还是会存在某些命题不可证。换句话说数学是永远发展不完的,所以直接破灭了希尔伯特的理想。我再举个例子吧,300多年前,许多数学家也是认为费马猜想是不可证的,但是现在还是得到了证明,就因为我们拥有了非常多的新公理和数学工具,伽罗瓦理论,椭圆曲线,代数几何,模函数,Hecke algebra,类域论等等使得费马猜想变成定理