雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(82)
It was Already well-known—it was known to Pythagoras—that there were irrational numbers.
The point of Cantor's construction was actually rather different from this.
但康託的重點並不在此，It was to show that no list could possibly contain all the 'real numbers', that is, all infinite decimals.
它說明的是，不可能有一個列表把所有的實數列出來，
For any proposed list would serve to define another infinite decimal which had been left out.
因爲任意舉出一個列表，都可以由它推出漏掉的數。Cantor's argument showed that in a quite precise sense there were more real numbers than integers.
康託精確地證明了實數比整數多，It opened up a precise theory of what was meant by 'infinite'.
還由此創立了一套精確的理論，來討論什麼是無限。However, the point relevant to Alan Turing's problem was that it showed how the rational could give rise to the irrational.
對艾倫來說，這個問題的意義是，它展示了怎樣由有理數推出無理數。In exactly the same way, therefore, the computable could give rise to the uncomputable, by means of a diagonal argument.
因此，用類似的方法，通過一個對角線證明，可計算也可以推出不可計算。As soon as he had made that observation, Alan could see that the answer to Hilbert's question was 'no'.
當艾倫想到這裏時，他馬上就知道了希爾伯特問題的答案——不。There could exist no 'definite method' for solving all mathematical questions.
不可能存在一種"機械的過程"來解決所有數學問題，For an uncomputable number would be an example of an unsolvable problem.
每一個不可計算數都是活生生的例子。There was still much work to do before his result was clear.
然而在他完全搞清楚之前，還存在很多工作要做。For one thing, there was something paradoxical about the argument.
一方面，這個論點看起來還有一點矛盾，The Cantor trick itself would seem to be a 'definite method'.
康託的對角線法本身，似乎就是一個機械的過程，The diagonal number was defined clearly enough, it appeared—so why could it not be computed?
對角線數是由明確的規則來生成的，爲什麼不可計算呢？How could something that was constructed in this mechanical way be uncomputable?
它是由機械的過程產生出來的，怎麼就不可計算了呢？What would go wrong, if it were attempted?
如果用機器來計算它，會出什麼問題呢
