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策梅洛写给哥德尔的信

已有 2169 次阅读 2022-5-5 16:18 |个人分类:解读哥德尔不完全性定理|系统分类:海外观察

19319月,恩斯特-策梅洛(Ernst Zermelo)写信给哥德尔,告知他认为哥德尔证明中存在一个“essential gap 重要的漏洞)Dawson, p. 76)。10月,哥德尔回了一封长达10页的信(Dawson,第76页,Grattan-Guinness,第512-513页),他指出策梅洛错误地假定一个系统中的真理概念在该系统中是可以定义的(根据塔斯基的不可定义性定理,这在一般情况下是不成立的)。但策梅洛不能接受,并将他的批评发表出来,对他的年轻竞争者有一段相当严厉的批评 ”Grattan-Guinness,第513页)。哥德尔决定,进一步追究此事毫无意义,卡尔纳普也同意(Dawson,第77页)。策梅洛后来的大部分工作都与比一阶逻辑更强的逻辑学有关,他希望用这些逻辑学来证明数学理论的一致性和范畴性。



一,哥德尔与策梅洛的通信概述 - JOHN W. DAWSON.JR. 


迄今为止,库尔特-哥德尔的科学通信很少被发表。一个突出的例外是1931年哥德尔和恩斯特-策梅洛之间的信件交流,这一交流始于921日策梅洛给哥德尔的一封信,批评不完全性定理。哥德尔于1012日回信,耐心地解释和辩护他的结果,但策梅洛仍未被说服;哥德尔1029日的回信终止了双方的通信。


这三封信中最后两封信的文本,是根据策梅洛在弗赖堡的Nuchlass的副本转录的,早些时候发表在本期刊上[Grattan Guinness 1979],以及Grattan-Guinness对通信的背景、内容和意义的评论;进一步的评论随后不久出现在[Moore 1980, 127-128]。然而,在策梅洛的论文中没有找到策梅洛921日的信,尽管根据哥德尔的答复可以推测出其内容(但不是其贬低的语气)。从逻辑上讲,我们可以在哥德尔的文件中找到这封信的副本,但不幸的是,Grattan-GuinnessMoore都是在哥德尔去世和整理他的Nachlass之间进行研究的。在整个这段时间里,接触哥德尔的论文受到严格限制,特别是Grattan-Guinness,被剥夺了在那里搜索的机会。后来,根据哥德尔在战争期间丢失了许多文件的报告, Grattan Guinness 推测策梅洛的第一封信一定是丢失了,而Moore则提到它显然没有幸存下来


然而,令人高兴的是,在我最近对Godel Nachlass进行编目的过程中,这封遗失的信件被发现了。我相信,它的确切措辞以及它所传达的态度,使得它值得在此发表,作为Moore文章的续篇和补充。因此,这里转载了这封四页纸的原始信件的照片,以及我自己的英文译文。


二,策梅洛写给哥德尔的信


亲爱的哥德尔先生:


我给你寄去了我的Fundumenta论文的证明样板,如果我可以把你算作少数几个至少试图接受那里的思想和方法并使他们在自己的研究中取得成果的人之一,我会很高兴。当我在准备我的Elster演讲的简短摘要时,在这个过程中我也不得不参考你的演讲,结果我清楚地认识到,你对不可判定命题的存在的证明出现了一个重要的漏洞essential gap)。为了产生一个不可判定的命题,你在第178页定义了一个类符号(一个自由变量的命题函数)S=R(q),然后你表明,无论是[R(q);q]=A还是其否定式¬A都是可证明的。但是


S =  Bew¬[R(n);n]


真的属于你的系统吗?你有理由把这个函数与R(q)相提并论,仅仅因为它是一个类符号?我知道以后会有关于类符号的详细理论,但为了进行批评,下面的考虑在这里就足够了:在你的公式(1)中,将符号组合“Bew”被省略,改为写成 


n in K* = ¬[R(n);n] = S*


如果再一次设定S*=R(q*),就会得出命题:


A* = R(q*; q*)


既不是也不是;也就是说,你的假设导致了一个类似于罗素悖论。正如在Richard悖论和Skolem悖论中一样,这个错误在于(错误的)假设,即每一个数学上可定义的概念都可以用有限的符号组合(根据一个固定的系统!)来表达我称之为有限性偏见。在现实中,情况是完全不同的,只有在这种偏见被克服之后(这是我的特殊职责),合理的元数学才有可能。正确地解释,你的证明思路,恰好将为此做出巨大贡献,从而为真理的事业提供实质性的服务。但就你现在的证明而言,我不能承认它有约束力。我想尽早把这一点传达给你,让你有时间来检查一下。


With best regards


E. Zermelo



三,Completing the Godel-Zermelo Correspondence 

JOHN W. DAWSON.JR.


Very little of Kurt Godel’s scientific correspondence has hitherto been published. An outstanding exception is the exchange of letters that took place during 1931 between Godel and Ernst Zermelo, an exchange that began on September 21 with a letter from Zermelo to Godel criticizing the incompleteness theorem. Godel replied on October 12, patiently explaining and defending his results, but Zermelo remained unconvinced; his response of October 29 terminated the correspondence. 


Texts of the last two of these three letters, transcribed from copies in Zermelo’s Nuchlass in Freiburg im Breisgau, were published earlier in this Journal [Grattan Guinness 19791, along with Grattan-Guinness’ commentary on the background, content, and significance of the correspondence; further commentary appeared shortly thereafter in [Moore 1980, 127-1281. Zermelo’s letter of September 21, however, was not found among Zermelo’s papers, though its content (but not its deprecatory tone) could be surmized on the basis of Godel’s reply. A copy of the letter might logically have been sought among Godel’s papers, but unfortunately, both Grattan-Guinness and Moore undertook their studies during the interim between Godel’s death and the organization of his Nachlass. Throughout that period, access to Godel’s papers was severely restricted, and Grattan-Guinness, in particular, was denied the opportunity to search there. Subsequently, on the basis of reports that Godel had lost many of his papers during the war, Grattan Guinness presumed that Zermelo’s first letter “must be lost,” while Moore in turn referred to it as having “apparently not survived.” 


Happily, however, in the course of my recent cataloguing of the Godel Nachlass the missing letter came to light. I believe that its exact wording, as well as the attitude it conveys, makes it worth publishing here as a sequel to and completion of Grattan-Guinness’ article. Accordingly, a photographic copy of the original four-page letter is reproduced here, together with my own English translation of the text. 


Dear Mr. Godel, 


I am sending you, enclosed, a proof-sheet of my Fundumenta paper, and I would be pleased if I might count you among the few who have at least tried to take up the ideas and methods developed there and make them fruitful for their own research. While I was engaged in preparing a short abstract of my Elster lecture, in the course of which I had also to refer to yours, I came subsequently to the clear realization that your proof of the existence of undecidable propositions exhibits an essential gap. In order to produce an “undecidable” proposition, you define on page 178 a “class sign” (a propositional function of one free variable) S = R(q), and then you show that neither [R(q);q] = A nor its negation ¬A would be “provable.” But does 


S =  Bew¬[R(n);n]


really belong to your “system,” and are you justified in identifying this function with R(q), just because it is a “class sign “? I know that later on there follows a detailed theory of “class signs,” but for a critique the following consideration suffices here: in your formula (1), let the sign combination “Bew” be omitted and write instead 


n in K* = ¬[R(n);n] = S*


If you then once more set S*=R(q*), it follows that the proposition


A* = R(q*; q*)


can be neither « true » nor « false »; that is, your assumption leads to a contradiction analogous to Russell’s antinomy. Just as in the Richard and Skolem paradoxes, the mistake rests on the (erroneous) assumption that every mathematically definable notion is expressible by a “finite combination of signs” (according to a fixed system!)-what I call the “finitistic prejudice.” In reality, the situation is quite different, and only after this prejudice has been overcome (a task I have made my particular duty) will a reasonable “metamathematics” be possible. Correctly interpreted, precisely your line of proof would contribute a great deal to this and could thereby render a substantial service to the cause of truth. But as your “proof” now stands, I cannot acknowledge it as binding. I wanted to impart this to you early on, to give you time to check it over. 


With best regards


E. Zermelo


参考文献:

1

NOTE Completing the Gbdel-Zermelo Correspondence

https://www.sciencedirect.com › science › article › pii › pdf

2https://en.wikipedia.org/wiki/Remarks_on_the_Foundations_of_Mathematics




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