雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(64)
Form now the assemblage of classes which are not members of themselves. This is a class: is it a member of itself or not?
If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself.
如果它屬於，那它就不是"不屬於自己的集合"，所以它不屬於；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.
但如果它不屬於，那它就是"不屬於自己的集合"，又應該屬於。
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.
無論它屬不屬於，都說不通，這就產生了矛盾。This paradox could not be resolved by asking what, if anything, it reAlly meant.
這個悖論，無論集合代表什麼，都是無法解決的。Philosophers could argue about that as long as they liked, but it was irrelevant to what Frege and Russell were trying to do.
哲學家們可以長期討論這個問題，愛多久就多久，但那些都與弗雷格和羅素要做的事情無關。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.
這個理論的關鍵，是要通過一種確定的、嚴密的、普適的、無爭議的方法，把算術問題從原始的邏輯中分離出來。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.
你不用關心羅素悖論代表什麼，它就是一組符號，這些符號本身，就能按照這個規則，無情地導致這個災難性的矛盾。In any purely logical system there was no room for a single inconsistency.
在任何一個純邏輯系統裏，都不能出現這樣的自相矛盾。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.
如果有人說2+2 = 5，那就能得出4=5，於是0=1，以至於任何數字都等於0，結果就是，任何等價於0＝0的命題，都是正確的。Mathematics, regarded in this game-like way, had to be totally consistent or it was nothing.
如果這樣看的話，數學要麼完全相容，要麼就全是浮雲。For ten years Russell and A.N. Whitehead laboured to remedy the defect.
在那十年中，羅素和A.N.懷特海，努力想要糾正這個錯誤。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'.
本質的困難是，現在已經證明，隨便弄一堆物體就叫作集合，這會導致自相矛盾。Some more refined definition was required.
我們需要更加精準的定義。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.
羅素悖論並不是集合論唯一的困境，但只有它在《數學原理》中佔了很大篇幅，這本1910年的權威著作，從原始邏輯中推導數學。The solution that Russell and Whitehead found was to set up a hierarchy of different kinds of sets, called 'types'.
羅素和懷特海提出的方法，是給不同的集合建立一套層次關係。There were to be primitive objects, then sets of objects, then sets of sets, then sets of sets of sets, and so on.
先有原始的對象，然後有對象的集合，然後又有集合的集合，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.
不同層次的集合，是不相同的，這樣一來，一個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，這又有了新的麻煩：本來想用這套理論來解釋數字系統，結果現在這套理論過於複雜，比數字系統本身還複雜。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.
不知道這是不是考慮集合和數字問題的唯一方法，在1930年，還有其它許多可供選擇的方案，其中，馮·諾伊曼也提出了一套。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.
數學應該是一個完備的相容的整體，這個聽起來不錯的需求，打開了一個充滿困難的潘多拉魔盒。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，數學命題看起來就像任何正確的東西一樣正確。但另一方面，它表現的只是紙上的符號，一旦有人糾纏符號的意義，這些符號就會引起悖論。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“愛麗絲鏡中奇遇”裏面的花園，你越是走向數學的心臟，就越會迷失在糾結的森林中。This lack of any simple connection between mathematical symbols and the world of actual objects fascinated Alan.
數學符號和物質實體之間沒有關聯，這個問題吸引了艾倫。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.
羅素在書的結尾說：“以上不完全的考量表明，在這個學科中，還有無數問題沒有解決，還有許多工作需要做。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.'
如果這本小書能夠給予學生啓發，對數理邏輯進行嚴肅的研究，那我寫這本書的主要目的就達到了。”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?
《數學原理》的主要目的確實達到了，因爲艾倫由此開始嚴肅地思考"層次"的問題——更大意義上說，他開始嚴肅地思考柏拉圖的問題：什麼是真理？
