Back for three months in the mild Cambridge summer of 1937, there were three major projects on hand.
1937年的平静的夏天，艾伦回到了剑桥，他手上有三个主要的项目。
First there was some tidying up of Computable Numbers.
首先是对《可计算数》进行一些整理。
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.
苏黎世的博纳斯，在他的证明中发现了一些错误，他必须在伦敦数学学会会刊上发表一个更正。
He also completed a formal demonstration that his own 'computability' coincided exactly with Church's 'effective calculability'.
他还要补充一个形式证明，表明他的"可计算"和丘奇的"有效过程"是等价的。
By now there existed yet a third definition of the same sort of idea.
现在有一个相前的定义是"递归函数"，
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;
这是一个非常巧妙的方式，可以用一些基本的函数来描述另外一个函数。
Gdel had suggested it, and it had been taken up by Kleene.
哥德尔提出了这个想法，而且被克林采用，
This idea, when formalised and extended somewhat, led to the definition of the 'recursive function'.
这个想法在哥德尔对不完备性的证明中起到了关键作用，并产生了"递归函数"的定义。
And now it had turned out that the general recursive function was exactly equivalent to the computable function.
现在可以证明，一般化的递归函数和可计算函数是等价的，
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算子和哥德尔的方法，都是与图灵机等价的。
Gdel himself later acknowledged the Turing machine construction as the most satisfactory definition of a 'mechanical procedure'.
而图灵机模型，是这些当中最接近机械过程的。