格哈德·根岑Gerhard Gentzen

格哈德·根岑的父亲是一名律师，在吕根岛上的卑尔根执业。正是在那里，根岑度过了他的童年时光，先上那里的elementary school，后来上Realgymnasium。然而，他的父亲在第一次世界大战中阵亡，1920年根岑的母亲搬到施特拉尔松德。根岑此时已经开始他的中学教育，但他在施特拉尔松德的Humanistische 文理中学（Gymnasium）继续他的教育。

当然，转学并没有影响根岑的学业成就，因为当他在1928年获得他的高中毕业考试（Abitur）时，是以优异成绩获得，并在学校排名第一。在[6]中，Robbel描述了年轻根岑的知识世界，特别考察了他的祖父母（尤其是A Bilharz）和父母对他的影响。他1928年Reifeprufung考试的结果在[6]的附录中给出。Humanistische Gymnasium的校长对结果印象深刻，并认识到他非凡的数学能力，授予他大学奖学金。

根岑，如当时通常情况，在不同德国大学之间流动。他于1928年在格赖夫斯瓦尔德大学开始数学学习，在那里学习两个学期后，他于1929年4月22日进入哥廷根大学。他再次只花了两个学期就继续前进，这次是去慕尼黑大学，在那里只花了一个学期，随后在柏林大学又花了一个学期。之后他回到哥廷根，在赫尔曼·外尔指导下攻读数学基础博士学位。他师从保罗·贝尔奈斯、康斯坦丁·卡拉西奥多里、理查·科朗特、大卫·希尔伯特、Kneser、埃德蒙·朗道，当然还有他的导师赫尔曼·外尔。

1933年，根岑在哥廷根获得博士学位，但在不同环境中的高强度研究已使他付出代价，因此在这一阶段他被迫回家休息、恢复健康。他回到哥廷根，于1934年成为大卫·希尔伯特的助手。M E Szabo在[2]中写道：-

……即使在大卫·希尔伯特退休后，他仍继续[在哥廷根]工作。在这些年里，根岑发表了一些他最重要的论文，并被赋予为《数学中央公报》审阅来自许多国家的众多著名研究者的著作这一重要任务。这些评论证明了他非凡的兴趣范围以及他在国际学者群体中的广泛参与。

正如我们已经提到的，根岑的工作是关于逻辑和数学基础的。他在1932年初将他的第一篇论文提交给Mathematische Annalen。该论文研究了“语句系统”理论，并通过构造一个反例表明并非所有语句系统都有独立公理系统，从而回答了该主题中的一个重大未决问题。然而，他也证明了线性语句系统确实有独立公理系统。他引入了“逻辑后承”的概念，这提供了一种比戈特洛布·弗雷格、伯特兰·罗素和大卫·希尔伯特提出的系统更接近数学推理的逻辑。这一思想后来被归功于阿尔弗雷德·塔斯基，他在1936年引入了它，比根岑晚三年。

1934年，根岑给出了简洁序列方法，即后承规则，这对推导元逻辑可判定性结果特别有用。大卫·希尔伯特让他研究公理方法以及将数学划分为层次。层次的观念大概最早由赫尔曼·外尔引入，它把数论视为第一层次，因为它处理自然数；把分析视为第二层次，因为它处理实数；把集合论视为第三层次，在那里将研究格奥尔格·康托尔的基数与序数的全部范围。根岑就这些概念写了几篇论文，尤其考察了集合论悖论的出现。

当然，库尔特·弗雷德里希·哥德尔发表他的不完备性定理正是在根岑开始其工作的时候。起初，根岑担心这会影响他在数学基础上想要实现的目标，并且由于担心库尔特·弗雷德里希·哥德尔定理的重要性，他在已经改正最终校样后撤回了本应成为他第二篇论文的文章。然而，后来他写到库尔特·弗雷德里希·哥德尔的结果时说：-

……这无疑是一个非常有趣、但肯定不令人担忧的结果。我们可以将其转述为：对于数论，无法指定一个一劳永逸的充分推理形式系统，相反，总能找到新的定理，其证明需要新的推理形式。

在1935年发表于Mathematische Zeitschrift的一篇论文中，根岑引入了谓词逻辑的两种新版本，现在称为$N$系统和$L$系统。次年，他为带归纳的算术系统S给出了一个基于$N$型逻辑的一致性证明。根岑在这篇论文的引言中写道：-

本文的目的是证明初等数论的一致性，或者更确切地说，是把一致性问题归约到某些基本原理上。

接着他探讨了为什么这样的—致性证明是必要的：——

数学被视为所有科学中最确定的一门。它竟会导致相互矛盾的结果，这似乎是不可能的。然而，这种对数学证明无可置疑的确定性的信念，在1900年前后因集合论中二律背反或悖论的发现而令人痛心地动摇了。结果表明，在数学的这一专门分支中，矛盾会出现，而我们却无法在我们的推理中识别出任何具体的错误。

在讨论了这些悖论，特别是伯特兰·罗素的悖论之后，根岑写道：——

……我将为初等数论实施这样一个一致性证明。然而，即使在这里，我们也会遇到一些推理形式，对它们的更仔细审视会给我们带来担忧的理由。……然而，有一点从一开始就应当明确：这些可能被认为有争议的推理形式，在实际的数论证明中几乎从不出现；我们切不可被误导，并且因为这些证明具有极大的自明性，就认为一致性证明是多余的。

根据库尔特·弗雷德里希·哥德尔的不可证明性定理，根岑所给出的这样一个证明必须使用比S更强的工具；根岑扩展了普通数学归纳法，使用了直到格奥尔格·康托尔的第一个epsilon数的超限归纳法，并且他还表明，这是此类证明所需的最低要求。

克林写道：-

……在何种程度上根岑的证明可以被接受为在那个问题表述的意义上确保了经典数论，这在目前情况下是一个个人判断的问题。

阿尔弗雷德·塔斯基写道：-

根岑关于算术一致性的证明无疑是一个非常有趣的元数学结果，可能被证明是非常有启发性和富有成果的。然而，我不能说算术的一致性现在对我来说比在证明给出之前更加明显……

根岑是对大卫·希尔伯特数学公理化纲领最杰出的贡献。1937年，他在巴黎的代表大会上作了题为Concept of infinity and the consistency of mathematics的报告。然而，他杰出的工作因第二次世界大战的爆发而中断。

根岑留在哥廷根任职直到1943年，尽管他在1939年至1941年间不得不服兵役。他被征召入伍，从事电信工作。然而他生病了，在军医院里恢复了三个月。他的健康状况太差，无法继续服兵役，于是回到了哥廷根。1942年夏天，他向哥廷根提交了教授资格论文（Habilitation）学位论文Provability and nonprovability of restricted transfinite induction in elementary number theory，获得学位后，他获得了在大学任教的资格。

作为德国战争努力的一部分，他在布拉格德国大学数学研究所担任Dozent教职，并在那里任教直到被捕并被拘留。布拉格市民于1945年5月5日反抗占领的德国军队，当天德国大学的所有教职员都被逮捕，并控制该城市直到四天后俄罗斯军队到达。人们必须提到Vihan在[8]中所述的关于根岑政治和军事生活的事实，即他与SA、NSDAP和NSD Dozentenbund的联系。根岑被俄罗斯军队拘留，关押条件恶劣。他在拘留3个月后死于营养不良。一位与他一起在监狱的朋友描述了他最后几天的情形：-

我能看到他躺在木床上，整天思考着那些占据他头脑的数学问题。他曾向我吐露，他其实相当满足，因为现在他终于有时间思考分析的相容性证明了……他还关心其他问题，比如人工语言等。他时不时会做一个简短的报告……我们不断得到安慰，说释放我们的手续只会再多花几天……他希望能回到哥廷根，全身心投入数学逻辑和数学基础的研究。他梦想为此建立一个研究所……

Gerhard Gentzen's father was a lawyer who practised law in Bergen on the Isle of Rügen. It was there that Gerhard spent his childhood years, attending first the elementary school there, and later the Realgymnasium. His father, however, was killed in World War I and in 1920 Gentzen's mother moved to Stralsund. Gentzen had already begun his secondary schooling at this stage but he continued his education at the Humanistische Gymnasium in Stralsund.

Certainly moving schools did not affect Gentzen's academic achievements for when he received his Abitur in 1928 it was with distinction and he was ranked top in his school. In [6] Robbel describes the intellectual world of the young Gentzen in particular examining the influences on him of his grandparents (especially A Bilharz) and his parents. The results of his 1928 Reifeprufung examination are given in an appendix to [6]. The headmaster of the Humanistische Gymnasium was certainly impressed with the results and, recognising his exceptional mathematical abilities, awarded him a university scholarship.

Gentzen, as was usual at this time, moved between different German universities. He began his mathematical studies at the University of Greifswald in 1928 then, after studying there for two semesters, he entered the University of Göttingen on 22 April 1929. Again he spent only two semesters before moving on, this time to the University of Munich where he spent only one semester followed by one further semester at the University of Berlin. After this he returned to Göttingen where he worked under Weyl for his doctorate on the foundations of mathematics. He was taught by Bernays, Carathéodory, Courant, Hilbert, Kneser, Edmund Landau and, of course, his supervisor Weyl.

In 1933 Gentzen was awarded his doctorate by Göttingen but the intense study in different environments had taken its toll so he was forced at this stage to return home to rest and recover his health. He returned to Göttingen, becoming Hilbert's assistant in 1934. M E Szabo writes in [2]:-

... he continued to work [at Göttingen] even after Hilbert's retirement. During these years Gentzen published some of his most important papers and was also given the responsible task of reviewing numerous works of eminent researchers from many countries for the Zentralblatt für Mathematik. These reviews attest his extraordinary range of interest and the great extent of his involvement in the international community of scholars.

As we have mentioned, Gentzen's work was on logic and the foundations of mathematics. He submitted his first paper to Mathematische Annalen early in 1932. The paper studies the theory of 'sentence systems' and answers a major open problem in the subject by constructing a counterexample to show that not all sentence systems have independent axiom systems. However he also showed that linear sentence systems do have independent axiom systems. He introduced the notion of 'logical consequence' which provided a logic closer to mathematical reasoning than the systems proposed by Frege, Russell and Hilbert. This idea was later attributed to Tarski who introduced it in 1936, three years after Gentzen.

In 1934 Gentzen gave the method of succinct Sequenzen, rules of consequents, which were particularly useful for deriving metalogical decidability results. Hilbert had him work on axiomatic methods and the classification of mathematics into levels. The idea of levels, probably first introduced by Weyl, considers number theory as the first level since it deals with the natural numbers, analysis as the second level since it deals with the real numbers, and set theory as the third level where the full extent of Cantor's cardinal and ordinal numbers would be studied. Gentzen wrote several papers on these concepts, particularly examining the occurrence of set theory paradoxes.

Of course Gödel published his incompleteness theorem just at the time Gentzen was beginning his work. At first Gentzen worried that it affected what he wanted to achieve on the foundations of mathematics and he withdrew what would have been his second paper after he had corrected the final proofs because of worries about the significance of Gödel's theorems. Later, however, he wrote of Gödel's result saying:-

... this is undoubtedly a very interesting, but certainly not an alarming, result. We can paraphrase it by saying that for number theory no once-and-for-all sufficient system of forms of inference can be specified, but that on the contrary, new theorems can always be found whose proof requires a new form of inference.

In a paper published in Mathematische Zeitschrift in 1935 Gentzen introduced two new versions of predicate logic now called the $N$-system and the $L$-system. In the following year he gave a consistency proof in terms of an $N$-type logic for the system S of arithmetic with induction. Gentzen wrote in the introduction to this paper:-

The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles.

He then looks at why such consistency proofs are necessary:-

Mathematics is regarded as the most certain of all sciences. That it could lead to results which contradict one another seems impossible. This faith in the indubitable certainty of mathematical proofs was sadly shaken around 1900 by the discovery of the antinomies or paradoxes of set theory. It turned out that in this specialised branch of mathematics, contradictions arise without our being able to recognise any specific error in our reasoning.

After discussing the paradoxes, in particular Russell's paradox, Gentzen writes:-

... I shall carry out such a consistency proof for elementary number theory. Yet even here we shall meet forms of inference whose closer inspection will give us cause for concern. ... One point should, however, be made clear from the outset: these forms of inference which might possibly be considered disputable hardly ever occur in actual number theoretical proofs; we must not be misled and, because of the great self-evidence of these proofs, consider a consistency proof as superfluous.

By Gödel's unprovability theorem, such a proof as Gentzen gave had to make use of tools stronger than those of S; extending ordinary mathematical induction, Gentzen employed transfinite induction up to Cantor's first epsilon number, and he also showed that this was the minimum required for such proof.

Kleene wrote:-

... to what extent the Gentzen proof can be accepted as securing classical number theory in the sense of that problem formulation is in the present state of affairs a matter of individual judgement.

Tarski wrote:-

Gentzen's proof of the consistency of arithmetic is undoubtedly a very interesting metamathematical result, which may prove very stimulating and fruitful. I cannot say, however, that the consistency of arithmetic is now much more evident to me ... than it was before the proof was given.

Gentzen's was the most outstanding contribution to Hilbert's programme of axiomatising mathematics. In 1937 he addressed the Congress in Paris giving a talk with title Concept of infinity and the consistency of mathematics. His outstanding work, however, was cut short by the start of World War II.

Gentzen remained on the staff at Göttingen until 1943, although he had to undertake military service in the years 1939 until 1941. He was conscripted into the army where he worked in telecommunications. He became ill, however, and spent three months recovering in a military hospital. His health was now too poor to allow him to continue with his military service and he returned to Göttingen. In the summer of 1942 he submitted his Habilitation thesis Provability and nonprovability of restricted transfinite induction in elementary number theory to Göttingen and, on the award of the degree, he became entitled to teach in universities.

As part of the German war effort, he took up a teaching post as a Dozent in the Mathematical Institute of the German University of Prague and he taught there until arrested and taken into custody. The citizens of Prague rose in revolt against the occupying German forces on 5 May 1945, the day all the staff of the German University were arrested, and held the city until the Russian Army arrived four days later. One would have to mention the facts concerning Gentzen's political and military life that Vihan relates in [8], namely his association with the SA, NSDAP and NSD Dozentenbund. Gentzen was interned by the Russian forces and held in poor conditions. He died of malnutrition after 3 months in internment. A friend who was in prison with him described his last few days:-

I can see him lying on his wooden bunk thinking all day about the mathematical problems that preoccupied him. He once confided in me that he was really quite content since now he had at last time to think about a consistency proof for analysis... He also concerned himself with other questions such as that of an artificial language, etc. Now and then he would give a short talk ... we were continually reassured that the formalities of our release would only take a few days longer.... he was hoping to be able to return to Göttingen and devote himself fully to the study of mathematical logic and the foundations of mathematics. He was dreaming of an Institute for this purpose ...