# 哥德尔给策梅洛的回信

19319月，策梅洛写给哥德尔的信中批评哥德尔的证明有误，哥德尔遂给策梅洛的回信。在Grattan-Guinness的这篇文章中有哥德尔的回信原文，因德文我无法翻译成中文，但通过Grattan-Guinness的转述，我们对哥德尔回信的内容可以略知一二。

1931年哥德尔发表了现在著名的不完全性定理后不久，哥德尔和策梅洛就哥德尔的结果的性质和意义进行了通信。在介绍现存信件的文本之前，先解释了通信的情况，并说明了所讨论的要点的历史意义。

IN MEMORIAM KURT GODEL:

HIS 1931 CORRESPONDENCE WITH ZERMELO ON HIS INCOMPLETABILITY THEOREM

BY I. GRATTAN-GUINNESS

MIDDLESEX POLYTECHNIC AT ENFIELD,

ENGLAND SUMMARIES

Shortly after publishing his now famous incompletability theorem in 1931, Kurt Godel and Ernst Zermelo corresponded about the nature and significance of Ggdel's result. The texts of the surviving letters are presented, preceded by an explanation of the circumstances of the correspondence and an indication of the historical significance of the points discussed.

INTRODUCTION

Kurt Gcdel (190671978) became a legend in his own life- time, principally for his theorem demonstrating the existence of a proposition which is expressible in the language of a formal system but which, like its negation, is not provable from the axioms of the system and its rules of inference. The result had major implications for two of the prevailing philosophies of mathematics (Hilbertian metamathematics and Russellian logicism), and has influenced many branches of logic ever since.

Godel proved his result in the year 1930. An abstract [Godel 1930b] was published at the end of the year, and the full paper [Godel 1931] appeared early in 1931. Later that year, on September 15, he gave a lecture on his theorem at the autumn meeting of the Deutsche Mathematiker-Vereinigung at Bad Elster. Zermelo also spoke at the same session, and later he wrote a report [Zermelo 1932a] on both his and Godel’s lectures. It was presumably in connection with this report that the correspondence with Godel occurred.

Zermelo must have written to Godel on September 21, 1931, for Godel acknowledged its receipt at the beginning of his 10-page reply of October 12 from Vienna. Zermelo's letter must be lost, for Godel lost many of his Vienna papers during the last war [see van Heijenoort 1967, 619]. But two carbon copies of Zermelo's answer of October 29 from Freiburg im Breisgau are preserved with Godel’s letter in Zennelo's Nachlass in the city's university library.

The main point of contention between the two men concerned a clash between their conceptions of proof, and the possibility of  the existence of undecidable propositions. In his paper of 1930 Zermelo had begun to use the axiom of regularity in his axiomatic set theory. The purpose of the axiom is to ‘well-foud’ sets by preventing unending descents of membership of sets, and Zermelo used it in [1932a] also to stratify the sets into mutually disjoint ‘layers’ [Schichten] (Qa) of sets. Very briefly, the idea is that Qo is the layer of ‘original elements’ of the axiomatic system, and each layer Qa is made up of those sets which do not belong to any preceding layers. The definition proceeds by transfinite induction : The index a starts at 0 and runs through a well-ordered series of ordinals. For limit ordinals the definition needs special treatment, whose details need not concern us here.

In a similar way Zermelo took a ‘basis’ of a mathematical theory to be composed of an ‘original range » of elements and ‘ground-relations’ between them. Propositions are then further relations built up by ‘quantification’ (meaning logical conjunction and disjonction) and negation of these relations. A collection of propositions is ‘well-founded’ on that basis with respect to the notion of consequence if each subcollection T of S contains at least one proposition which is not a consequent of any member of S.

He showed that S could be stratified into ‘layers of quantification’, each layer indexed by an ordinal. He then argued (rather vaguely!) that if the ground-relations are divided arbitrarily into ‘true’ and ‘false’, then the division carried over into each layer of propositions, and that if the ground-relations are consistent (surely he means those which are designed as true, although then their designation cannot be arbitrary), then the derived relations are also consistent. For him it then followed that any proposition which is true is also provable in the sense of belonging to some level of quantification. But this situation would, naturally, be contradicted by Godel’s theorem.

In response to Zermelo’s lost letter of September 21, Godel  carefully explains the workings of his proof. He begins by confirming that the first section of his paper  [1931] is not a ‘binding proof’ of his theorem, but a guide to the correctness of the proof which then follows in great detail in the paper, based on the 46 definitions. He then mentions the similarity between his proof-method and Cantor’s diagonal argument (as he does also in this opening part of his paper) : This is a feature which will arise later. But first on pages 2)3 he discusses the class of ‘true’ formulae which Zermelo seems to have proposed in the hope of generating a paradox like the liar paradox in Godel’s system. Godel shows in detail on page 3-7 that, unlike his own class of provable formule, Zermelo’s class of true formulae cannot ne expressed in a ‘purely combinatorial’ way in the language of his system, so that no such paradox will arise.

In these pages Godel also lays emphasis on the metamathematical character of the argument; that is metamathematical we talk about names of formulae and so on Despite the attention given to the distinction between theory and me theory in the 1920s with Hilbert’s revival of his proof theory, logicians still tended to conflate symbol and referent; indeed, Professor J. Barkley Rosser once told me in reminiscence that it was only with Godel’s theorem that logicians realised how carful they needed to be in this matter.

Godel then goes on (pages 7-8) to tree that the class of provable formulae must in fact be strictly contained in the class of true formulae, since his theorem exhibits a true but unprovable formula. He makes the interesting comment that his proof is not free from criticism from intuitionists. He then points out on pages 8-9 the bearing of Cantor’s diagonal argument : not so much that one can extend beyond any formal system with new formulae (though Cantor’s argument does show that), but that there are propositions which can ‘express themselves’ within a system but cannot be decided within it, and that such systems can be of such a simple kind as the one that he constructed. He draws on Cantor’s diagonal argument again to show that all mathematics cannot be drawn into a single system, but points out that his theorem shows that even the fundamental component of mathematics which he has treated cannot be made complete. He recalls that in his paper he stated that undecidable propositions can be made decidable in higher systems, but that his theorem showed that further undecidable propositions would be found there, and so on.

Godel’s letter finished on pages 9-10 with pleasantries, including a readiness to comment on Zermelo’s paper of 1930. Unfortunately for us, Zermelo’s reply on October 29 did not accept the invitation but concentrated on Godel’s last point, the possibility of decidability in higher system. He stressed that these higher systems may have new proof methods in them as well as new propositions and drew on Cantor’s diagonal argument to characterise Godel’s theorem as confining proof ‘finitistically’ to a denumerable subset of provable propositions. There may be more general systems - and doubtless he had in mind his own scheme - where transfinite ordinals were used in the indexing of the levels of quantifications.

Although Zermelo reworked his ideas in [1932b] and [1935], his proposal is too vague to have been influential on the later attempts to extend proofs to infinitely lengths. (Such later work is surveyed in, for example, [Karp 1964]). Zermelo seems not to have fully understood the implications of Godel’s result. I see here an example of an historical principle which I have noticed in action many times before : not merely that contemporaries of a new result or approach have difficulties with it which later generations master easily, but especially that new comers to that result or approach do go through some of the same difficulties that concerned those contemporaries. It is a principle expressible by the slogan ‘education imitates history,’ and is one of the reasons why I deplore the non historical character of mathematical education. Zermelo’s comments on Godel’s theorem, and Godel’s replies, are very similar to the conversations which one has in the classroom with students trying to grasp the signification of one of the most remarkable intellectual discoveries of this century.

1In memoriam Kurt Godel: His 1931 correspondence with zermelo on his incompletability theorem

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