The answer was that one could not tell.
这个问题的答案是，判断不了。
There was no way of checking in advance that a table would produce an infinite sequence.
没有办法提前检查一个表能不能产生一个无限序列。
There might be a method for some particular table.
也许有办法检查某些特定的表，
But there was no mechanical process—no machine—that could work on all instruction tables.
但没有一个机械的过程——没有一个机器，能够检查所有的指令表。
There was nothing better than the prescription: 'take the table and try it out'.
我们顶多只能说：运行那个表试一下。
But this procedure would take an infinite time to find out whether infinitely many digits emerged.
但是可以想见，要想试验能否产生无限序列，这就需要无限的时间。
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.
没有一种规则能够在有限的时间里，检查任意的行为表，正如对角线数不能在有限的时间内打印出来。
The Cantor process, therefore, could not be mechanised, and the uncomputable diagonal number could not be computed.
所以，康托的对角线法，不能机械化，不可计算的对角线数，确实不可计算。
There was no paradox after all.
现在，一点矛盾也没有了。
Alan called the description numbers which gave rise to infinite decimals the 'satisfactory numbers'.
如果一个描述数，能够产生无限小数，艾伦把它称为可用数。
So he had shown that there was no definite method of identifying an 'unsatisfactory number'.
于是他表明，没有明确的方法能识别出一个不可用数。
He had pinned down a clearly specified example of something Hilbert said did not exist—an unsolvable problem.
他用一个非常明显的例子证明，希尔伯特说的那种东西是不存在的。
There were other ways of demonstrating that no 'mechanical process' could eliminate the unsatisfactory numbers.
还有一些其它方法也能表明，不存在任何机械过程，能够筛选不可用数。
The one he himself favoured was one which brought out the connection with self-reference in the question.
他自己最喜欢的方法是，这个问题中包含了自我指涉。
For supposing that such a 'checking' machine did exist, able to locate the unsatisfactory numbers, it could be applied to itself.
假如存在这样的机器，能够检查不可用数，那它也可以检查它自己。
But this, he showed, led to a flat contradiction. So no such checking machine could exist.
然后他证明，这会导致自相矛盾，所以不存在这样的机器。
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.
无论哪种方法都能证明，这个问题是不可解的。艾伦现在只需要一个技术性的步骤，就能用严格的形式解决希尔伯特的问题。