雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(66)

In the same way, extending the axioms of arithmetic could be done by one infinite list of axioms, or by two, or by infinitely many infinite lists, there was again no limit.

同理，也可以用這種方法，把一個無限長的算術公理表，擴展成兩倍，甚至無限多倍的無限列表，也是無限的。

The question was whether any of this would overcome the Gdel effect.

現在的問題是，是否存在一個這樣的列表，使哥德爾定理不適用。

Cantor had described his different orderings of the integers by 'ordinAl numbers', and Alan described his different extensions of the axioms of arithmetic as 'ordinal logics'.

康託用序數來描述他的這些整數序列，而艾倫則把擴展算術公理系統稱爲序數邏輯。

In one sense it was clear that no 'ordinal logic' could be 'complete', in Hilbert's technical sense.

某種意義上講，很明顯，在希爾伯特的角度來看，這些序數邏輯都是不完備的。

For if there were infinitely many axioms, they could not all be written out.

因爲如果有無限多個的公理，我們就無法把它們全寫出來，

There would have to be some finite rule for generating them.

所以必須要有一組有限的公理來生成它們。

But in that case, the whole system would still be based on a finite number of rules, so Gdel's theorem would still apply to show that there were still unprovable assertions.

這樣一來，這個公理系統還是基於有限的規則，所以哥德爾定理仍然適用，也就是說，其中仍然存在無法證明的命題。

However, there was a more subtle question.

然而，還有一個更加難以捉摸的問題。

In his 'ordinal logics', the rule for generating the axioms was given in terms of substituting an 'ordinal formula' into a certain expression.

在艾倫的序數邏輯中，一個產生公理的規則，是把一個序數公式代入一個特定的表達式。

This was itself a 'mechanical process'.

這是一個機械的過程。

But it was not a 'mechanical process' to decide whether a given formula was an ordinal formula.

但是，判斷一個公式是不是序數公式，並不是一個機械過程。