前几日看完了Godel, Escher, Bach一书，其中对哥德尔不完备性定理证明的叙述是我见过的最精彩的（因为我只读过这一个证明-_-）。这个证明是核心，书里其他部分也非常之有意思。

然后想找来哥德尔原论文对照一下，网上可以搜到这个英译德的文章。翻译成了我们现在习惯的符号。证明部分13页，有兴趣的童鞋看看吧，可以看懂的。熟悉数学证明形式的童鞋，看起来会觉得很自然。当然看完GEB后再看这个，理解起来要轻松一些。
文章链接：
http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf

另有一篇图灵的讨论机器能否思考的论文，即关于人工智能的东西。文中提出图灵测验，我觉得可以取名叫伪装测验，对应于我国优秀的传统文学中的真假孙悟空的段子。详情参考原文。最后反驳了几个“机器不能像人一样思考”的观点，其中有一观点是作为哥德尔定理的“推论”。

(3) The Mathematical Objection

There are a number of results of mathematical logic which can be used to show that there are limitations to the powers of discrete-state machines. The best known of these results is known as Godel's theorem ( 1931 ) and shows that in any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent. There are other, in some respects similar, results due to Church (1936), Kleene (1935), Rosser, and Turing (1937). The latter result is the most convenient to consider, since it refers directly to machines, whereas the others can only be used in a comparatively indirect argument: for instance if Godel's theorem is to be used we need in addition to have some means of describing logical systems in terms of machines, and machines in terms of logical systems. The result in question refers to a type of machine which is essentially a digital computer with an infinite capacity. It states that there are certain things that such a machine cannot do. If it is rigged up to give answers to questions as in the imitation game, there will be some questions to which it will either give a wrong answer, or fail to give an answer at all however much time is allowed for a reply. There may, of course, be many such questions, and questions which cannot be answered by one machine may be satisfactorily answered by another. We are of course supposing for the present that the questions are of the kind to which an answer "Yes" or "No" is appropriate, rather than questions such as "What do you think of Picasso?" The questions that we know the machines must fail on are of this type, "Consider the machine specified as follows. . . . Will this machine ever answer 'Yes' to any question?" The dots are to be replaced by a description of some machine in a standard form, which could be something like that used in §5. When the machine described bears a certain comparatively simple relation to the machine which is under interrogation, it can be shown that the answer is either wrong or not forthcoming. This is the mathematical result: it is argued that it proves a disability of machines to which the human intellect is not subject.

The short answer to this argument is that although it is established that there are limitations to the Powers If any particular machine, it has only been stated, without any sort of proof, that no such limitations apply to the human intellect. But I do not think this view can be dismissed quite so lightly. Whenever one of these machines is asked the appropriate critical question, and gives a definite answer, we know that this answer must be wrong, and this gives us a certain feeling of superiority. Is this feeling illusory? It is no doubt quite genuine, but I do not think too much importance should be attached to it. We too often give wrong answers to questions ourselves to be justified in being very pleased at such evidence of fallibility on the part of the machines. Further, our superiority can only be felt on such an occasion in relation to the one machine over which we have scored our petty triumph. There would be no question of triumphing simultaneously over all machines. In short, then, there might be men cleverer than any given machine, but then again there might be other machines cleverer again, and so on.

文章链接：
http://loebner.net/Prizef/TuringArticle.html
值得一读。