This idea enabled one-ness to be defined in terms of same-ness, or equality.

这样就可以用相等的概念来定义一个。

But then equality could be defined in terms of satisfying the same range of predicates.

同时，相等还可以定义为对任意谓词有同样的值域。

In this way the concept of number and the axioms of arithmetic could, it appeared, be rigorously derived from the most primitive notions of entities, predicates and propositions.

这样来看的话，数字概念和算术公理就可以通过最原始的实体、谓词和命题而严格地推导出来。

Unfortunately it was not so simple.

不幸的是，事情并没有这么简单。

Russell wanted to define a set-with-one-element, without appealing to a concept of counting, by the idea of equality.

罗素希望不通过计数，而是通过相等的概念，来定义单元素集合，

Then he would define the number ‘one’ to be ‘the set of all sets-with-one-element’.

然后再用包含所有单元素集合的集合来定义数字1。

But in 1901 Russell noticed that logical contradictions arose as soon as one tried to use ‘sets of all sets’.

但是在1901年，罗素发现，这种集合的集合会引发逻辑矛盾。

The difficulty arose through the possibility of self-referring, self-contradictory assertions, such as ‘this statement is a lie.’ One problem of this kind had emerged in the theory of the infinite developed by the German mathematician G. Cantor.

这个问题就在于，自我指涉的结果，有可能导致自相矛盾，比如这句话是谎言。在德国数学家G.康托的无限理论中，也出现了类似的问题，

Russell noticed that Cantor’s paradox had an analogy in the theory of sets.

罗素发现，康托悖论和集合论悖论是很类似的。

He divided the sets into two kinds, those that contained themselves, and those that did not.

他把集合分成两种，一类包含自己，一类不包含自己。

‘Normally’, wrote Russell, ‘a class is not a member of itself.

罗素写道：一般来说，集合不是自己的一个元素，比如人类的一个元素是一个人，但人类本身不是一个人。

Mankind, for example, is not a man.’ But the set of abstract concepts, or the set of all sets, would contain itself.

然而，如果考虑抽象概念的集合，或者集合的集合，它就有可能是自己的一个元素。

Russell then explained the resulting paradox in this way:

罗素接着说，这就有可能引发悖论：