It was a staggering thought, since this list would include every number that could be arrived at through arithmetical operations, finding roots of equations, and using mathematical functions like sines and logarithms—every number that could possibly arise in computational mathematics.
这是一个非常了不起的想法，因为这个列表包括了任何可以通过算术运算得到的数，比如方程求解，或者正弦和对数这样的数学函数。
And once he had seen this, he knew the answer to Hilbert's question.
当艾伦意识到这一点，他就知道了希尔伯特的问题的答案，
Probably it was this that he suddenly saw on the Grantchester meadow.
这就是他在格兰彻斯特的草地上突然发现的奥秘。
He would have seen the answer because there was a beautiful mathematical device, ready to be taken off the shelf.
现在有一个很漂亮的数学工具，磨拳擦掌，准备要出场了。
Fifty years earlier, Cantor had realised that he could put all the fractions—all the ratios or rational numbers—into a list.
早在50年前康托就发现，他可以把所有的分数——所有的比值或者说有理数——放进一个列表。
Naively it might be thought that there were many more fractions than integers.
如果从直觉上考虑，小数似乎比整数多很多。
But Cantor showed that, in a precise sense, this was not so, for they could be counted off, and put into a sort of alphabetical order.
但是康托展示了，如果严格地看，并不是这样的，因为它们是可数的，并且可以按照某种顺序排列。
Omitting fractions with cancelling factors, this list of all the rational numbers between 0 and 1 would begin:
我们只考虑约分后的分数，那么0到1之间的有理数就可以表示成：
1/2 1/3 1/4 2/3 1/5 1/6 2/5 3/4 1/7 3/5 1/8 2/7 4/5 1/9 3/7 1/10…
1/2 1/3 1/4 2/3 1/5 1/6 2/5 3/4 1/7 3/5 1/8 2/7 4/5 1/9 3/7 1/10...
Cantor went on to invent a certain trick, called the Cantor diagonal argument, which could be used as a proof that there existed irrational numbers.
接着，康托继续展示一种技巧，叫作康托对角线证明，来证明存在无理数。
For this, the rational numbers would be expressed as infinite decimals, and the list of all such numbers between 0 and 1 would then begin:
首先用无限小数来表示有理数，于是得到一个0到1之间的这样的数的列表：
5000000000000000000.…
1 .5000000000000000000
3333333333333333333.…
2 .3333333333333333333
2500000000000000000.…
3 .2500000000000000000
6666666666666666666.…
4 .6666666666666666666
2000000000000000000.…
5 .2000000000000000000. .. .
1666666666666666666.…
6 .1666666666666666666
4000000000000000000.…
7 .4000000000000000000. ...
7500000000000000000.…
8 .7500000000000000000
1428571428571428571.…
9 .1428571428571428571
6000000000000000000.…
10 .6000000000000000000
1250000000000000000.…
11 .1250000000000000000
2857142857142857142.…
12 .2857142857142857142....
8000000000000000000.…
13 .8000000000000000000
1111111111111111111.…
14 .1111111111111111111....
4285714285714285714.…
15 .4285714285714285714
1000000000000000000.…
16 .1000000000000000000....
The trick was to consider the diagonal number, beginning
这个技巧就是考虑对角线上的数，也就是：
.5306060020040180.…
.5306060020040180……
and then to change each digit, as for instance by increasing each by 1 except by changing a 9 to a 0. This would give an infinite decimal beginning
然后改变其中的每个数字，比如每一位都加1, 9改成0，那么就得到一个新的无限小数：
.6417171131151291.…
.6417171131151291……
a number which could not possibly be rational, since it would differ from the first listed rational number in the first decimal place, from the 694th rational number in the 694th decimal place, and so forth. Therefore it could not be in the list;
这个数不可能是有理数，因为它的第1位与表中第1个数的第1位不同，它的第694位与表中第694个数的第649位不同，以此类推，它与表中的每个数都不同，所以它不在这个列表中。
but the list held all the rational numbers, so the diagonal number could not be rational.
但是因为这个列表包括了所有的有理数，所以这个对角线数不是有理数。