Form now the assemblage of classes which are not members of themselves. This is a class: is it a member of itself or not?
考虑一个集合，它的元素是所有的“不属于自己的集合”，那这个集合本身属不属于它自己？
If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself.
如果它属于，那它就不是"不属于自己的集合"，所以它不属于；
If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself.
但如果它不属于，那它就是"不属于自己的集合"，又应该属于。
Thus of the two hypotheses—that it is, and that it is not, a member of itself —each implies its contradictory. This is a contradiction.
无论它属不属于，都说不通，这就产生了矛盾。
This paradox could not be resolved by asking what, if anything, it really meant.
这个悖论，无论集合代表什么，都是无法解决的。
Philosophers could argue about that as long as they liked, but it was irrelevant to what Frege and Russell were trying to do.
哲学家们可以长期讨论这个问题，爱多久就多久，但那些都与弗雷格和罗素要做的事情无关。
The whole point of this theory was to derive arithmetic from the most primitive logical ideas in an automatic, watertight, depersonalised way, without any arguments en route.
这个理论的关键，是要通过一种确定的、严密的、普适的、无争议的方法，把算术问题从原始的逻辑中分离出来。
Regardless of what Russell's paradox meant, it was a string of symbols which, according to the rules of the game, would lead inexorably to its own contradiction. And that spelt disaster.
你不用关心罗素悖论代表什么，它就是一组符号，这些符号本身，就能按照这个规则，无情地导致这个灾难性的矛盾。
In any purely logical system there was no room for a single inconsistency.
在任何一个纯逻辑系统里，都不能出现这样的自相矛盾。
If one could ever arrive at '2 + 2 = 5' then it would follow that '4 = 5', and '0 = 1', so that any number was equal to 0, and so that every proposition whatever was equivalent to '0 = 0' and therefore true.
如果有人说2+2 = 5，那就能得出4=5，于是0=1，以至于任何数字都等于0，结果就是，任何等价于0＝0的命题，都是正确的。
Mathematics, regarded in this game-like way, had to be totally consistent or it was nothing.
如果这样看的话，数学要么完全相容，要么就全是浮云。
For ten years Russell and A.N. Whitehead laboured to remedy the defect.
在那十年中，罗素和A.N.怀特海，努力想要纠正这个错误。
The essential difficulty was that it had proved self-contradictory to assume that any kind of lumping together of objects could be called 'a set'.
本质的困难是，现在已经证明，随便弄一堆物体就叫作集合，这会导致自相矛盾。
Some more refined definition was required.
我们需要更加精准的定义。
The Russell paradox was by no means the only problem with the theory of sets, but it alone consumed a large part of Principia Mathematical, the weighty volumes which in 1910 set out their derivation of mathematics from primitive logic.
罗素悖论并不是集合论唯一的困境，但只有它在《数学原理》中占了很大篇幅，这本1910年的权威著作，从原始逻辑中推导数学。
The solution that Russell and Whitehead found was to set up a hierarchy of different kinds of sets, called 'types'.
罗素和怀特海提出的方法，是给不同的集合建立一套层次关系。
There were to be primitive objects, then sets of objects, then sets of sets, then sets of sets of sets, and so on.
先有原始的对象，然后有对象的集合，然后又有集合的集合，集合的集合的集合，等等。
By segregating the different 'types' of set, it was made impossible for a set to contain itself.
不同层次的集合，是不相同的，这样一来，一个集合就不可能包含它自己。
But this made the theory very complicated, much more difficult than the number system it was supposed to justify.
但是，这又有了新的麻烦：本来想用这套理论来解释数字系统，结果现在这套理论过于复杂，比数字系统本身还复杂。
It was not clear that this was the only possible way in which to think about sets and numbers, and by 1930 various alternative schemes had been developed, one of them by von Neumann.
不知道这是不是考虑集合和数字问题的唯一方法，在1930年，还有其它许多可供选择的方案，其中，冯·诺伊曼也提出了一套。
The innocuous-sounding demand that there should be some demonstration that mathematics formed a complete and consistent whole had opened a Pandora's box of problems.
数学应该是一个完备的相容的整体，这个听起来不错的需求，打开了一个充满困难的潘多拉魔盒。
In one sense, mathematical propositions still seemed as true as anything could possibly be true; in another, they appeared as no more than marks on paper, which led to mind-stretching paradoxes when one tried to elucidate what they meant.
一方面，数学命题看起来就像任何正确的东西一样正确。但另一方面，它表现的只是纸上的符号，一旦有人纠缠符号的意义，这些符号就会引起悖论。
As in the Looking-Glass garden, an approach towards the heart of mathematics was liable to lead away into a forest of tangled technicalities.
正如“爱丽丝镜中奇遇”里面的花园，你越是走向数学的心脏，就越会迷失在纠结的森林中。
This lack of any simple connection between mathematical symbols and the world of actual objects fascinated Alan.
数学符号和物质实体之间没有关联，这个问题吸引了艾伦。
Russell had ended his book saying, 'As the above hasty survey must have made evident, there are innumerable unsolved problems in the subject, and much work needs to be done.
罗素在书的结尾说：“以上不完全的考量表明，在这个学科中，还有无数问题没有解决，还有许多工作需要做。
If any student is led into a serious study of mathematical logic by this little book, it will have served the chief purpose for which it has been written.'
如果这本小书能够给予学生启发，对数理逻辑进行严肃的研究，那我写这本书的主要目的就达到了。”
So the Introduction to Mathematical Philosophy did serve its purpose, for Alan thought seriously about the problem of 'types' — and more generally, faced Pilate's question: What is truth?
《数学原理》的主要目的确实达到了，因为艾伦由此开始严肃地思考"层次"的问题——更大意义上说，他开始严肃地思考柏拉图的问题：什么是真理？