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，也可以用这种方法，把一个无限长的算术公理表，扩展成两倍，甚至无限多倍的无限列表，也是无限的。
The question was whether any of this would overcome the Gdel effect.
现在的问题是，是否存在一个这样的列表，使哥德尔定理不适用。
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'.
康托用序数来描述他的这些整数序列，而艾伦则把扩展算术公理系统称为序数逻辑。
In one sense it was clear that no 'ordinal logic' could be 'complete', in Hilbert's technical sense.
某种意义上讲，很明显，在希尔伯特的角度来看，这些序数逻辑都是不完备的。
For if there were infinitely many axioms, they could not all be written out.
因为如果有无限多个的公理，我们就无法把它们全写出来，
There would have to be some finite rule for generating them.
所以必须要有一组有限的公理来生成它们。
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.
这样一来，这个公理系统还是基于有限的规则，所以哥德尔定理仍然适用，也就是说，其中仍然存在无法证明的命题。
However, there was a more subtle question.
然而，还有一个更加难以捉摸的问题。
In his 'ordinal logics', the rule for generating the axioms was given in terms of substituting an 'ordinal formula' into a certain expression.
在艾伦的序数逻辑中，一个产生公理的规则，是把一个序数公式代入一个特定的表达式。
This was itself a 'mechanical process'.
这是一个机械的过程。
But it was not a 'mechanical process' to decide whether a given formula was an ordinal formula.
但是，判断一个公式是不是序数公式，并不是一个机械过程。