罗素回应哥德尔定理（译文）

3. 罗素回应哥德尔定理

• 我们两个人都不可能再写这本书了；即使在一起，并通过相互讨论带来了缓解，情况是如此严重，以至于最后我们都带着一种厌恶从数理逻辑转向其他 [26, p. 138]

• Dr. G¨odel’s most interesting paper on my mathematical logic came into my hands after my replies had been completed, and at a time when I had no leisure to work on it. As it is now about eighteen years since I last worked on mathematical logic, it would have taken me a long time to form a critical estimate of Dr. G¨odel’s opinions. His great ability, as shown in his previous work, makes me think it highly probable that many of his criticisms of me are justiﬁed. The writing of Principia Mathematica was completed thirty-three years ago, and obviously, in view of subsequent advances in the subject, it needs amending in various ways. If I had the leisure, I should be glad to attempt a revision of its introductory portions, but external circumstances make this impossible. I must therefore ask the reader to give Dr. G¨odel’s work the attention that it deserves, and to form his own critical judgment on it [35, p. 741].

• 哥德尔博士关于我的数理逻辑的最有趣的论文是在我的答复完成后，在我没有闲暇时间工作的时候到我手中的。由于距离我上次研究数理逻辑已经过去了18年，我需要花很长时间才能对哥德尔博士的观点形成一个批判性的评估。他在以前的工作中表现出的巨大能力，使我认为他对我的许多批评很可能是有道理的。数学原理的写作是在33年前完成的，很明显，鉴于该学科后来的进展，它需要在各方面进行修正。如果我有闲暇，我应该很乐意尝试对其介绍性部分进行修订，但外部环境使我无法做到。因此，我必须请读者对哥德尔博士的作品给予应有的关注，并形成自己的批评性判断[35, p. 741]

• I have received G¨odel’s article, and a letter from him, urging me to answer it. It is quite impossible for me to make a detailed answer. I have not worked at mathematical logic since 1927 and it would take me at least a month’s work. I am prepared to write a short paragraph saying I am unable to form a critical estimate of his article, but I think it quite probable that most of his criticisms are justiﬁed. I hope this will satisfy him and you [13, p. 231].

• 我收到了哥德尔的文章，以及他的一封信，敦促我回答。我不可能做出详细的回答。我自1927年以来就没有从事过数理逻辑方面的工作，这至少要花费我一个月的时间。我准备写一小段话，说我无法对他的文章形成批评性的评价，但我认为他的大部分批评很可能是有道理的。我希望这能让他和你满意[13, p. 231]

1945年发表的一篇关于逻辑实证主义的文章中，罗素明确地将哥德尔不完备性定理描述为一个悖论:

• Carnap has shown that it is possible for a language to say things about its own syntax, but there always remain things which can not be said in the original language, but only in the meta-language....The development of logical syntax on these lines, especially by Carnap, is very elaborate and technically dicult. Nor can it be said, as yet, to have reached a deﬁnitive form. A new set of paradoxes has been discovered by G¨odel, and there can be no security that there are not others to follow [25].

• 卡纳普已经表明，一种语言有可能对它自己的语法说些什么，但总是有一些东西不能在原始语言中说出来，而只能在元语言中说出来….。按照这些思路对逻辑语法的发展，特别是卡纳普的发展，是非常复杂的，在技术上也很困难。目前也不能说它已经达到了一个确定的形式。哥德尔已经发现了一组新的悖论，而且不能保证后面不会有其他悖论[25]

• A new set of puzzles has resulted from the work of G¨odel, especially his article ¨ Uber formal unentscheidbare S¨ atze der Principia Mathematica und verwandter Systeme  (1931), in which he proved that in any formal system it is possible to construct sentences of which the truth or falsehood cannot be decided within the system. Here again we are faced with the essential necessity of a hierarchy, extending upwards ad inﬁnitum, and logically incapable of completion

• 哥德尔的工作带来了一系列新的难题，特别是他的文章《Uber formal unentscheidbare S¨ atze der Principia Math-ematica und verwandter Systeme》（1931年），其中他证明在任何形式系统中，都有可能构造出真假无法在系统内判定的句子。在这里，我们再次面临着层次的基本必要性，它是向上不停延伸，而且在逻辑上无法完成。

• Thank you very much for your letter of March 26 and for the very interesting paper which you enclosed. I have read the latter carefully and it has given me much new information. It is ﬁfty years sinceI worked seriously at mathematical logic and almost the only work that I have read since that date is G¨odel’s. I realized, of course, that G¨odel’s work is of fundamental importance, but I was puzzled by it. It made me glad that I was no longer working at mathematical logic. If a given set of axioms leads to a contradiction, it is clear that at least one of the axioms must be false. Does this apply to school-boys’ arithmetic, and, if so, can we believe anything that we were taught in youth? Are we to think that 2+2 is not 4, but 4.001? Obviously, this is not what is intended [14, p. 592].

• 非常感谢你326日的信和你所附的非常有趣的文件。我仔细阅读了后者，它给了我很多新的信息。自从我认真研究数理逻辑以来，已经有很多年了，从那时起，我读过的几乎唯一的作品就是哥德尔的。当然，我意识到哥德尔的工作具有根本的重要性，但我对它感到困惑不解。这让我庆幸自己不再从事数理逻辑研究。如果一组给定的公理导致了矛盾，那么很明显，至少有一个公理必须是假的。这是否适用于小学生的算术，如果是这样，我们能相信年轻时被教导的任何东西吗？难道我们要认为2+2不是4，而是4.001？显然，这不是我们的目的[14, p. 592]

• Kreisel had said that it was astonishing how acute Russell’s under-standing had seemed to be even though he had not done any work on mathematical logic for about thirty years. It was not merely that his brain was beautifully clear for somebody of 87; it was beautifully clear for anybody of any age. In particular, Kreisel had told Russell about some new developments in connection with the notion of eectiveness – the one developed by Turing. Russell had clearly not been very familiar with the notion before, but he had immediately been able to follow all its complications and implications [3, pp. 129-130].

• Kreisel曾说，令人惊讶的是，罗素的理解力似乎非常敏锐，即使他没有做任何工作的数学逻辑约30年。这不仅仅是他的大脑是美丽清晰的87岁的人;这是美丽清晰的任何人的任何年龄。特别是，Kreisel告诉罗素一些新的发展，与eectiveness的概念有关 - 一个由图灵开发。罗素显然没有非常熟悉的概念之前，但他立即能够跟进所有其复杂和含义[3，第129-130]

Irving Anellis FOM中报告说，罗素的信（及其 "愚人节 "日期）促使亨金在198315日至9日在科罗拉多州丹佛举行的美国数学学会年会的证明理论特别会议上问他，罗素是否在开玩笑。Anellis的意见是，这封信的整个基调，连同罗素得出结论说哥德尔的结果允许小学生算术教学有2 + 2 = 4.001的哲学背景，表明罗素真的是认真的。

1973年，莱昂-亨金（Leon Henkin）在亚伯拉罕-罗宾逊（Abraham Robinson）的推动下，将罗素的信的副本寄给了库尔特-哥德尔。在回复罗宾逊早先的一封信时，G¨odel对罗素的言论评论如下：

• Russell evidently misinterprets my result; however, he does so in a very interesting manner, which has a bearing on some of the questions we discussed a few months ago. In contradistinction Wittgen-stein, in his posthumous book, advances a completely trivial and uninteresting misinterpretation [13, p. 201].

• 罗素显然误解了我的结果；然而，他是以一种非常有趣的方式这样做的，这对我们几个月前讨论的一些问题有影响。与此相反，维特根斯坦在他的遗著中提出了一个完全微不足道且无趣的误读[13，第201]

• Not long after the appearance of Principia Mathematica, G¨odel propounded a new diculty. He proved that, in any systematic logical language, there are propositions which can be stated, but cannot be either proved or disproved. This has been taken by many (not, I think, by G¨odel) as a fatal objection to mathematical logic in the form which I and others had given to it. I have never been able to adopt this view. It is maintained by those who hold this view that no systematic logical theory can be true of everything. Oddly enough, they never apply this opinion to elementary everyday arithmetic. Until they do so, I consider that they may be ignored. I had always supposed that there are propositions in mathematical logic which can be stated, but neither proved nor disproved. Two of these had a fairly prominent place in Principia Mathematica – namely, the axiom of choice and the axiom of inﬁnity. To many mathematical logicians, however, the destructive inﬂuence of G¨odel’s work appears much greater than it does to me and has been thought to requirea great restriction in the scope of mathematical logic. ... I adhere to the view that one should make the best set of axioms that one can think of and believe in it unless and until actual contradictions appear [31, p. xviii].

• 在《数学原理》出现后不久，哥德尔提出了一个新的困难。他证明，在任何系统的逻辑语言中，有些命题可以被陈述，但不能被证明或证伪。这被许多人（我想不是由哥德尔）认为是对我和其他人提出的数学逻辑的致命反对。我从来没能采纳这种观点。持这种观点的人认为，没有一个系统逻辑理论可以对一切都是真的。奇怪的是，他们从未将这一观点应用于初级的日常算术。在他们这样做之前，我认为他们可以被忽略。我一直认为，数理逻辑中有些命题是可以陈述的，但既不能证明也不能证伪。其中有两个命题在《数学原理》中占有相当重要的地位，即选择公理和无穷公理。然而，对许多数理逻辑学家来说，哥德尔的工作的破坏性影响似乎比对我的破坏性影响大得多，并被认为需要对数理逻辑的范围进行极大的限制。... 我坚持这样的观点：一个人应该制定他能想到的最好的公理集，并相信它，除非出现实际的矛盾[31, p. xviii]

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