雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(72)
He then showed that all the operations of 'proof', these 'chess-like' rules of logical deduction, were themselves arithmetical in nature.
That is, they would only employ such operations as counting and comparing, in order to test whether one expression had been correctly substituted for another—just as to see whether a chess move was legal or not would only be a matter of counting and comparing.
也就是說，它們只是通過計數和比較這種操作，來判斷一個命題是否能被另一個命題替代――就像判斷棋子的移動是否合法，只是計數和比較而已。In fact, Gdel showed that the formulae of his system could be encoded as integers, so that he had integers representing statements about integers. This was the key idea.
實際上，哥德爾表明，可以對這個系統進行編碼，這樣就可以用數字來表示關於數字的命題。這是他的核心想法。
Gdel continued to show how to encode proofs as integers, so that he had a whole theory of arithmetic, encoded within arithmetic.
哥德爾繼續展示，如何把證明編碼，以便整個算術系統都能用算術的方式描述。It was an exploitation of the fact that if mathematics were regarded purely as a game with symbols, then it might as well employ numerical symbols as any other.
這個擴展基於這個事實：如果數學是一個純粹的符號遊戲，那就可以把符號全部換成數字。He was able to show that the property of 'being a proof' or of 'being provable' was no more and no less arithmetical than the property of 'being square' or 'being prime'.
他能夠說明，“是一個證明”或“是可證明的”這樣的性質，跟“是平方數”或“是素數”一樣算術化。The effect of this encoding process was that it became possible to write down arithmetical statements which referred to themselves, like the person saying 'I am lying.'
這個編碼的結果是，人們能夠寫出自我指涉的算術命題，比如那個人說“我這句是說謊”。Indeed Gdel constructed one particular assertion which had just such a property, for in effect it said 'This statement is unprovable.'
哥德爾確實構建了一個具有這樣的性質的命題，他說，“這個命題是不可證明的”。It followed that this assertion could not be proved true, for that would lead to a contradiction. Nor could it be proved false, for the same reason.
這個命題既無法證明，也無法證僞，因爲它會導致自相矛盾。It was an assertion which could not be proved or disproved by logical deduction from the axioms, and so Gdel had proved that arithmetic was incomplete, in Hilbert's technical sense.
一個命題用公理進行邏輯推演，卻既不能證明，也不能證僞，所以，對於希爾伯特的問題來說，哥德爾已經證明，算術是不完備的。There was more to it than this, for one remarkable thing about Gdel's special assertion was that since it was not provable, it was, in a sense, true.
還有，哥德爾的特殊命題還有一個明顯的問題。因爲它是不可證明的，所以從某種意義上來說，它永遠是真的。But to say it was 'true' required an observer who could, as it were, look at the system from outside.
但如果要說它是"真"的，就需要一個外部的觀察者，從這個系統之外來看待。It could not be shown by working within the axiomatic system.
你不能在這個公理系統內部來表明這一結論。Another point was that the argument assumed that arithmetic was consistent.
另外一點是，這個論點假設了算術是相容的。If, in fact, arithmetic were inconsistent, then every assertion would be provable.
實際上，如果算術不是相容的，那麼每個命題都可以被證明。So more precisely, Gdel had shown that formalised arithmetic must either be inconsistent, or incomplete.
所以更確切地，哥德爾表明，一個形式算術系統，要麼不完備，要麼不相容。He was also able to show that arithmetic could not be proved consistent within its own axiomatic system.
他也能夠說明，算術在它自身的公理系統中，可以證明是相容的。To do so, all that would be required would be a proof that there was a single proposition (say, 2 + 2 = 5) which could not be proved true.
要做到這一點，需要這樣一個證明：存在一個不能被證明爲“真”的命題（比如2 + 2 = 5）。But Gdel was able to show that such a statement of existence had the same character as the sentence that asserted its own unprovability.
哥德爾能夠說明，這樣的命題，與宣佈自己不可證明的句子，本質上是一樣的。And in this way, he had polished off the first two of Hilbert's questions.
這樣一來，他解決了希爾伯特的前兩個問題。Arithmetic could not be proved consistent, and it was certainly not consistent and complete.
算術無法被證明是相容的，而且一定不是既完備又相容的。This was an amazing new turn in the enquiry, for Hilbert had thought of his programme as one of tidying up loose ends.
這是數學發展中的一個驚人的轉折，因爲希爾伯特已經認爲，他的計劃已經準備收尾了。It was upsetting for those who wanted to find in mathematics something that was absolutely perfect and unassailable; and it meant that new questions came into view.
這使那些想要在數學中找到絕對完美的人們感到沮喪，它意味着，有新的重大問題出現了
