He then showed that all the operations of 'proof', these 'chess-like' rules of logical deduction, were themselves arithmetical in nature.
接着他展示了，所有的证明，那些像国际象棋一样的逻辑演算规则，它们自己本质上就是算术的。
That is, they would only employ such operations as counting and comparing, in order to test whether one expression had been correctly substituted for another—just as to see whether a chess move was legal or not would only be a matter of counting and comparing.
也就是说，它们只是通过计数和比较这种操作，来判断一个命题是否能被另一个命题替代――就像判断棋子的移动是否合法，只是计数和比较而已。
In fact, Gdel showed that the formulae of his system could be encoded as integers, so that he had integers representing statements about integers. This was the key idea.
实际上，哥德尔表明，可以对这个系统进行编码，这样就可以用数字来表示关于数字的命题。这是他的核心想法。
Gdel continued to show how to encode proofs as integers, so that he had a whole theory of arithmetic, encoded within arithmetic.
哥德尔继续展示，如何把证明编码，以便整个算术系统都能用算术的方式描述。
It was an exploitation of the fact that if mathematics were regarded purely as a game with symbols, then it might as well employ numerical symbols as any other.
这个扩展基于这个事实：如果数学是一个纯粹的符号游戏，那就可以把符号全部换成数字。
He was able to show that the property of 'being a proof' or of 'being provable' was no more and no less arithmetical than the property of 'being square' or 'being prime'.
他能够说明，“是一个证明”或“是可证明的”这样的性质，跟“是平方数”或“是素数”一样算术化。
The effect of this encoding process was that it became possible to write down arithmetical statements which referred to themselves, like the person saying 'I am lying.'
这个编码的结果是，人们能够写出自我指涉的算术命题，比如那个人说“我这句是说谎”。
Indeed Gdel constructed one particular assertion which had just such a property, for in effect it said 'This statement is unprovable.'
哥德尔确实构建了一个具有这样的性质的命题，他说，“这个命题是不可证明的”。
It followed that this assertion could not be proved true, for that would lead to a contradiction. Nor could it be proved false, for the same reason.
这个命题既无法证明，也无法证伪，因为它会导致自相矛盾。
It was an assertion which could not be proved or disproved by logical deduction from the axioms, and so Gdel had proved that arithmetic was incomplete, in Hilbert's technical sense.
一个命题用公理进行逻辑推演，却既不能证明，也不能证伪，所以，对于希尔伯特的问题来说，哥德尔已经证明，算术是不完备的。
There was more to it than this, for one remarkable thing about Gdel's special assertion was that since it was not provable, it was, in a sense, true.
还有，哥德尔的特殊命题还有一个明显的问题。因为它是不可证明的，所以从某种意义上来说，它永远是真的。
But to say it was 'true' required an observer who could, as it were, look at the system from outside.
但如果要说它是"真"的，就需要一个外部的观察者，从这个系统之外来看待。
It could not be shown by working within the axiomatic system.
你不能在这个公理系统内部来表明这一结论。
Another point was that the argument assumed that arithmetic was consistent.
另外一点是，这个论点假设了算术是相容的。
If, in fact, arithmetic were inconsistent, then every assertion would be provable.
实际上，如果算术不是相容的，那么每个命题都可以被证明。
So more precisely, Gdel had shown that formalised arithmetic must either be inconsistent, or incomplete.
所以更确切地，哥德尔表明，一个形式算术系统，要么不完备，要么不相容。
He was also able to show that arithmetic could not be proved consistent within its own axiomatic system.
他也能够说明，算术在它自身的公理系统中，可以证明是相容的。
To do so, all that would be required would be a proof that there was a single proposition (say, 2 + 2 = 5) which could not be proved true.
要做到这一点，需要这样一个证明：存在一个不能被证明为“真”的命题（比如2 + 2 = 5）。
But Gdel was able to show that such a statement of existence had the same character as the sentence that asserted its own unprovability.
哥德尔能够说明，这样的命题，与宣布自己不可证明的句子，本质上是一样的。
And in this way, he had polished off the first two of Hilbert's questions.
这样一来，他解决了希尔伯特的前两个问题。
Arithmetic could not be proved consistent, and it was certainly not consistent and complete.
算术无法被证明是相容的，而且一定不是既完备又相容的。
This was an amazing new turn in the enquiry, for Hilbert had thought of his programme as one of tidying up loose ends.
这是数学发展中的一个惊人的转折，因为希尔伯特已经认为，他的计划已经准备收尾了。
It was upsetting for those who wanted to find in mathematics something that was absolutely perfect and unassailable; and it meant that new questions came into view.
这使那些想要在数学中找到绝对完美的人们感到沮丧，它意味着，有新的重大问题出现了