雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(47)
Back for three months in the mild Cambridge summer of 1937, there were three major projects on hand.
First there was some tidying up of Computable Numbers.
首先是對《可計算數》進行一些整理。Bernays, in Zurich, had perhaps rather annoyingly found some errors13 in his proof that the Hilbert decision problem, in its precise form, was unsolvable, and these had to be put right by a correction note in the LMS Proceedings.
蘇黎世的博納斯，在他的證明中發現了一些錯誤，他必須在倫敦數學學會會刊上發表一個更正。
He also completed a formal demonstration that his own 'computability' coincided exactly with Church's 'effective calculability'.
他還要補充一個形式證明，表明他的"可計算"和丘奇的"有效過程"是等價的。By now there existed yet a third definition of the same sort of idea.
現在有一個相前的定義是"遞歸函數"，This was the 'recursive function', which was a way of making absolutely precise the notion of defining a mathematical function in terms of other more elementary functions;
這是一個非常巧妙的方式，可以用一些基本的函數來描述另外一個函數。Gdel had suggested it, and it had been taken up by Kleene.
哥德爾提出了這個想法，而且被克林採用，This idea, when formalised and extended somewhat, led to the definition of the 'recursive function'.
這個想法在哥德爾對不完備性的證明中起到了關鍵作用，併產生了"遞歸函數"的定義。And now it had turned out that the general recursive function was exactly equivalent to the computable function.
現在可以證明，一般化的遞歸函數和可計算函數是等價的，So Church's lambda-calculus, and Gdel's way of defining arithmetical functions, both turned out to be equivalent to the Turing machine.
所以丘奇的λ算子和哥德爾的方法，都是與圖靈機等價的。Gdel himself later acknowledged the Turing machine construction as the most satisfactory definition of a 'mechanical procedure'.
而圖靈機模型，是這些當中最接近機械過程的。
