雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(80)
But there was an approach which led to the answer by a back door route. Alan hit on the idea of the 'computable numbers'.
The crucial notion was that any 'real number' which was defined by some definite rule could be calculated by one of his machines.
這個關鍵的想法是，任意一個由明確規則定義的實數，都可以用一個這樣的機器來計算出來。For instance, there would be a machine to calculate the decimal expansion of π, rather as he had done at school.
比如說，存在一個機器，來計算圓周率π，就像他在學校時人工算的那樣。
For it would require no More than a set of rules for adding, multiplying, copying, and so forth.
因爲這隻需要一套加、乘、複製的規則。being an infinite decimal, the work of the machine would never end, and it would need an unlimited amount of working space on its 'tape'.
因爲它是一個無限小數，所以這個機器將永遠不會停止，而且它需要無限長的紙帶。But it would arrive at every decimal place in some finite time, having used only a finite quantity of tape.
但在某個特定的時刻，它會處於某一個小數位，並且只用了有限的紙帶。And everything about the process could be defined by a finite table, left alone to work on a blank tape.
整個計算過程都可以用行爲表來規定，然後把它丟在那裏，讓它獨自在紙帶上跑來跑去This meant that he had a way of representing a number like π, an infinite decimal, by a finite table.
這就是說，他現在得到了一種方法，可以用有限的表格，來表示無限的小數，比如π。The same would be true of the square root of three, or the logarithm of seven—or any other number defined by some rule.
對於3的平方根或7的對數也可以，任何一個由規則定義的數字都可以。Such numbers he called the 'computable numbers'.
他稱這樣的數爲“可計算數”。More precisely, the machine itself would know nothing about decimals or decimal places.
準確地說，機器本身對小數或小數位一無所知，It would simply produce a sequence of digits.
它只是產生一串數字序列。A sequence that could be produced by one of his machines, starting on a blank tape, he called a 'computable sequence'.
他的一個機器，從一條空白的紙帶開始，產生這樣的序列，他稱爲“可計算序列”。Then an infinite computable sequence, prefaced by a decimal point, would define a 'computable number' between 0 and 1.
然後，用一個以小數點作爲開始的可計算序列，就可以定義一個0到1之間可計算數。It was in this more strict sense that any computable number between 0 and 1 could be defined by a finite table.
嚴格來說，任意0到1之間的可計算數，都可以用有限的行爲表來定義。It was important to his argument that the computable numbers would always be expressed as infinite sequences of digits, even if these were all 0 after a certain point.
他的論點有一個重要之處：任何可計算數字，總是由一個無限的序列來表示，哪怕它每一位都是0。But these finite tables could now be put into something like alphabetical order, beginning at the most simple, and working through larger and larger ones.
現在我們來考慮這些行爲表，從簡單的開始，到越來越複雜的，They could be put in a list, or counted; and this meant that all the computable numbers could be put in a list.
它們本身也可以按某種順序排列起來，成爲一個列表。這就意味着，所有的可計算數可以構成一個列表。It was not a practical proposition actually to do it, but in principle the idea was perfectly definite, and would result in the square root of three being say 678th in order, or the logarithm of π being 9369th.
雖然在現實中不太可能真的寫出這個列表，但這個想法本身是可以完美定義的，這樣一來，3的平方根可能是第678個，而77的對數可能是第9369個。
