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It was already well-known—it was known to Pythagoras—that there were irrational numbers.
对于无理数的存在，毕达哥拉斯早就搞清楚了。
The point of Cantor's construction was actually rather different from this.
但康托的重点并不在此，
It was to show that no list could possibly contain all the 'real numbers', that is, all infinite decimals.
它说明的是，不可能有一个列表把所有的实数列出来，
For any proposed list would serve to define another infinite decimal which had been left out.
因为任意举出一个列表，都可以由它推出漏掉的数。
Cantor's argument showed that in a quite precise sense there were more real numbers than integers.
康托精确地证明了实数比整数多，
It opened up a precise theory of what was meant by 'infinite'.
还由此创立了一套精确的理论，来讨论什么是无限。
However, the point relevant to Alan Turing's problem was that it showed how the rational could give rise to the irrational.
对艾伦来说，这个问题的意义是，它展示了怎样由有理数推出无理数。
In exactly the same way, therefore, the computable could give rise to the uncomputable, by means of a diagonal argument.
因此，用类似的方法，通过一个对角线证明，可计算也可以推出不可计算。
As soon as he had made that observation, Alan could see that the answer to Hilbert's question was 'no'.
当艾伦想到这里时，他马上就知道了希尔伯特问题的答案——不。
There could exist no 'definite method' for solving all mathematical questions.
不可能存在一种"机械的过程"来解决所有数学问题，
For an uncomputable number would be an example of an unsolvable problem.
每一个不可计算数都是活生生的例子。
There was still much work to do before his result was clear.
然而在他完全搞清楚之前，还存在很多工作要做。
For one thing, there was something paradoxical about the argument.
一方面，这个论点看起来还有一点矛盾，
The Cantor trick itself would seem to be a 'definite method'.
康托的对角线法本身，似乎就是一个机械的过程，
The diagonal number was defined clearly enough, it appeared—so why could it not be computed?
对角线数是由明确的规则来生成的，为什么不可计算呢？
How could something that was constructed in this mechanical way be uncomputable?
它是由机械的过程产生出来的，怎么就不可计算了呢？
What would go wrong, if it were attempted?
如果用机器来计算它，会出什么问题呢
