雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(81)

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.

這是一個非常了不起的想法，因爲這個列表包括了任何可以通過算術運算得到的數，比如方程求解，或者正弦和對數這樣的數學函數。

And once he had seen this, he knew the answer to Hilbert's question.

當艾倫意識到這一點，他就知道了希爾伯特的問題的答案，

Probably it was this that he suddenly saw on the Grantchester meadow.

這就是他在格蘭徹斯特的草地上突然發現的奧祕。

He would have seen the answer because there was a beautiful mathematical device, ready to be taken off the shelf.

現在有一個很漂亮的數學工具，磨拳擦掌，準備要出場了。

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年前康託就發現，他可以把所有的分數——所有的比值或者說有理數——放進一個列表。

Naively it might be thought that there were many more fractions than integers.

如果從直覺上考慮，小數似乎比整數多很多。

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.

但是康託展示了，如果嚴格地看，並不是這樣的，因爲它們是可數的，並且可以按照某種順序排列。

Omitting fractions with cancelling factors, this list of all the rational numbers between 0 and 1 would begin:

我們只考慮約分後的分數，那麼0到1之間的有理數就可以表示成：

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，康託繼續展示一種技巧，叫作康託對角線證明，來證明存在無理數。

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:

首先用無限小數來表示有理數，於是得到一個0到1之間的這樣的數的列表：

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

這個技巧就是考慮對角線上的數，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

然後改變其中的每個數字，比如每一位都加1, 9改成0，那麼就得到一個新的無限小數：

.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;

這個數不可能是有理數，因爲它的第1位與表中第1個數的第1位不同，它的第694位與表中第694個數的第649位不同，以此類推，它與表中的每個數都不同，所以它不在這個列表中。

but the list held all the rational numbers, so the diagonal number could not be rational.

但是因爲這個列表包括了所有的有理數，所以這個對角線數不是有理數。