雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(67)
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'.
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.
如果這一步能成功，可能就會得到一個完備的系統，其中的任何命題都可證明的，只是沒有一個機械的過程來描述這個公理系統是什麼。He likened the job of deciding whether a formula was an ordinal formula to 'intuition'.
艾倫把判定一個公式是否是序數公式的這項工作，比作是一種直覺。
In a 'complete ordinal logic', any theorem in arithmetic could be proved by a mixture of mechanical reasoning, and steps of 'intuition'.
在一個完備的序數邏輯中，任何算術定理都能夠通過機械過程配合這種直覺來證明。In this way, he hoped to bring the Gdel incompleteness under some kind of control.
艾倫希望通過這種方式，使哥德爾定理的力量得到一定的控制。But he regarded his results as disappointingly negative.
但是很遺憾，他的結論是消極的。'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.
完備的邏輯確實存在，但是有一個問題，人們無法知道在證明一個定理的過程中，有多少個步驟要依靠直覺。There was no way of measuring how 'deep' a theorem was, in his sense; no way of pinning down exactly what was going on.
用艾倫的話說："我們無法衡量一個定理有多『深』，也說不清楚這個系統在做什麼。"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).
在這個問題上，艾倫有個想法，他想到一種算卦式的圖靈機，一個這種機器對應着一個不可解的問題（比如判定一個序數公式），This introduced the idea of relative computability, or relative unsolvability, which opened up a new field of enquiry in mathematical logic.
這就引入了"相對可計算性"的觀點，或者說相對不可計算性，於是開創了一個數理邏輯的新領域。
