What he asked was whether all the incompleteness of arithmetic could be concentrated in one place, namely into the unsolvable problem of deciding which formulae were 'ordinal formulae'.
艾伦的问题是，能否把算术系统中所有的不完备性集中起来，集中到一个不能解决的问题上，这个问题就是判定哪些公式是序数公式。
If this could be done, then there would be a sense in which arithmetic was complete; everything could be proved from the axioms, although there would be no mechanical way of saying what the axioms were.
如果这一步能成功，可能就会得到一个完备的系统，其中的任何命题都可证明的，只是没有一个机械的过程来描述这个公理系统是什么。
He likened the job of deciding whether a formula was an ordinal formula to 'intuition'.
艾伦把判定一个公式是否是序数公式的这项工作，比作是一种直觉。
In a 'complete ordinal logic', any theorem in arithmetic could be proved by a mixture of mechanical reasoning, and steps of 'intuition'.
在一个完备的序数逻辑中，任何算术定理都能够通过机械过程配合这种直觉来证明。
In this way, he hoped to bring the Gdel incompleteness under some kind of control.
艾伦希望通过这种方式，使哥德尔定理的力量得到一定的控制。
But he regarded his results as disappointingly negative.
但是很遗憾，他的结论是消极的。
'Complete logics' did exist, but they suffered from the defect that one could not count the number of 'intuitive' steps that were necessary to prove any particular theorem.
完备的逻辑确实存在，但是有一个问题，人们无法知道在证明一个定理的过程中，有多少个步骤要依靠直觉。
There was no way of measuring how 'deep' a theorem was, in his sense; no way of pinning down exactly what was going on.
用艾伦的话说："我们无法衡量一个定理有多『深』，也说不清楚这个系统在做什么。"
One nice touch on the side was his idea of an 'oracle' Turing machine, one which would have the property of being able to answer one particular unsolvable problem (like recognising an ordinal formula).
在这个问题上，艾伦有个想法，他想到一种算卦式的图灵机，一个这种机器对应着一个不可解的问题（比如判定一个序数公式），
This introduced the idea of relative computability, or relative unsolvability, which opened up a new field of enquiry in mathematical logic.
这就引入了"相对可计算性"的观点，或者说相对不可计算性，于是开创了一个数理逻辑的新领域。