雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(65)
The fundamental idea was to add further axioms to the system, in such a way that the 'true but unprovable' statements could be proved.
It was easy enough to add an axiom so that one of Gdel's peculiar statements could be proved.
添加一條公理是很容易的，可以使一條命題變得可證明，But then Gdel's theorem could be applied to the enlarged set of axioms, producing yet another 'true but unprovable' assertion.
但問題是，哥德爾定理同樣適用於擴大後的公理集，所以又會產生新的不可證明的命題。
It could not be enough to add a finite number of axioms; it was necessary to discuss adding infinitely many.
因此，加入有限多個公理是不夠的，必須要討論另一種情形，那就是加入無限多個公理。This was just the beginning, for as mathematicians well knew, there were many possible ways of doing 'infinitely many' things in order.
這僅僅是個開始，數學家們都知道，有很多方法可以處理無限問題。Cantor had seen this when investigating the notion of ordering the integers.
康託在研究整數的次序時，就考慮到了這一點。Suppose, for example, that the integers were ordered in the following way:
他假設說，如果把整數這樣排列：first all the even numbers, in ascending order, and then all the odd numbers.
首先是所有的偶數，按升序排列，然後是所有的奇數。In a precise sense, this listing of the integers would be 'twice as long' as the usual one.
從直覺上來說，這樣排成的序列，應該是正常順序的兩倍長，It could be made three times as long, or indeed infinitely many times as long, by taking first the even numbers, then remaining multiples of 3, then remaining multiples of 5, then remaining multiples of 7, and so on.
同理還可以排成三倍長，甚至可以是無限倍長。Indeed, there was no limit to the 'length' of such lists.
總之，這個序列的長度是無限的。
