雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(85)
The answer was that one could not tell.
There was no way of checking in advance that a table would produce an infinite sequence.
沒有辦法提前檢查一個表能不能產生一個無限序列。There might be a method for some particular table.
也許有辦法檢查某些特定的表，
But there was no mechanical process—no machine—that could work on all instruction tables.
但沒有一個機械的過程——沒有一個機器，能夠檢查所有的指令表。There was nothing better than the prescription: 'take the table and try it out'.
我們頂多只能說：運行那個表試一下。But this procedure would take an infinite time to find out whether infinitely many digits emerged.
但是可以想見，要想試驗能否產生無限序列，這就需要無限的時間。There was no rule that could be applied to any table, and be guaranteed to produce the answer in a finite time, as was required for the printing of the diagonal number.
沒有一種規則能夠在有限的時間裏，檢查任意的行爲表，正如對角線數不能在有限的時間內打印出來。The Cantor process, therefore, could not be mechanised, and the uncomputable diagonal number could not be computed.
所以，康託的對角線法，不能機械化，不可計算的對角線數，確實不可計算。There was no paradox after all.
現在，一點矛盾也沒有了。Alan called the description numbers which gave rise to infinite decimals the 'satisfactory numbers'.
如果一個描述數，能夠產生無限小數，艾倫把它稱爲可用數。So he had shown that there was no definite method of identifying an 'unsatisfactory number'.
於是他表明，沒有明確的方法能識別出一個不可用數。He had pinned down a clearly specified example of something Hilbert said did not exist—an unsolvable problem.
他用一個非常明顯的例子證明，希爾伯特說的那種東西是不存在的。There were other ways of demonstrating that no 'mechanical process' could eliminate the unsatisfactory numbers.
還有一些其它方法也能表明，不存在任何機械過程，能夠篩選不可用數。The one he himself favoured was one which brought out the connection with self-reference in the question.
他自己最喜歡的方法是，這個問題中包含了自我指涉。For supposing that such a 'checking' machine did exist, able to locate the unsatisfactory numbers, it could be applied to itself.
假如存在這樣的機器，能夠檢查不可用數，那它也可以檢查它自己。But this, he showed, led to a flat contradiction. So no such checking machine could exist.
然後他證明，這會導致自相矛盾，所以不存在這樣的機器。Either way, he had found an unsolvable problem, and it required only a technical step to show that this settled Hilbert's question about mathematics, in the exact form in which it had been posed.
無論哪種方法都能證明，這個問題是不可解的。艾倫現在只需要一個技術性的步驟，就能用嚴格的形式解決希爾伯特的問題。
