亚伯拉罕·弗兰克尔Abraham Fraenkel

亚伯拉罕·弗兰克尔的父母是Sigmund Aviezri 弗兰克尔（1860-1925）及其妻子Charlotte Chaja Sara Neuburger（1868-1965），两人都出生在巴伐利亚的慕尼黑。Sigmund是一名羊毛商人，他与Charlotte于1889年11月3日在慕尼黑结婚。我们在此应当指出，这个家庭是犹太家庭，事实上Sigmund除了是商人之外，还是巴伐利亚正统犹太教的领袖，以及为德国政府工作的经济学家。本传记的主人公马克斯·亚伯拉罕，直到晚年前往巴勒斯坦之前一直被称为弗兰克尔，从那时起他自称亚伯拉罕。然而，为避免混淆，我们在整篇传记中将称他为亚伯拉罕。他是父母子女中的长子，有弟弟Joel Eugen（生于1892年3月25日）、妹妹Else Lea（生于1896年7月11日）和妹妹Paula Tirza（生于1901年5月29日）。

弗兰克尔五岁时开始接受教育，当时他的父母为这个年幼的孩子聘请了一位私人教师。他从这位私人教师那里学习希伯来语，然后进入慕尼黑的一所小学。他进入了Luitpold 文理中学（Gymnasium），这所学校由巴伐利亚的Luitpold亲王于1891年创立，正是弗兰克尔出生的那一年。这所文理中学服务于慕尼黑东部，位于Alexandrastrasse，对面是国家博物馆。弗兰克尔是一名出色的学生，他所学的每一门科目都获得了“非常好”的成绩。他于1909年7月从文理中学毕业。他已经撰写了关于宗教历法系统的文章：Bestimmung des Datums des jüdischen Osterfestes für die Zeitrechnung der Mohammedaner Ⓣ（确定穆斯林时代犹太逾越节的日期）（1908年）；Eine Formel zur Verwandlung jüdischer Daten in mohammedanische Ⓣ（一种在犹太历和穆斯林历之间转换的公式）（1909年）；以及Die Berechnung des Osterfestes Ⓣ（复活节的确定）（1910年）。这些文章结合了他对宗教和数学的兴趣。

与他那个时代德国的大多数学生一样，弗兰克尔曾在不同的大学学习过一段时间。他在慕尼黑大学度过了一个学期，然后前往马尔堡大学，在那里听了库尔特·亨泽尔、Ernst Richard Neumann（1875-1955）和恩斯特·黑林格的课程。随后他前往柏林弗里德里希-威廉大学（自1949年起称为柏林洪堡大学），在那里受到赫尔曼·阿曼杜斯·施瓦茨、费迪南德·格奥尔格·弗罗贝尼乌斯和弗里德里希·赫曼·肖特基的教导。他在布雷斯劳大学度过了最后一年的学习，然后回到马尔堡大学，在库尔特·亨泽尔的指导下提交了他的博士论文Über die Teiler der Null und die Zerlegung von Ringen Ⓣ（论零因子与环的分解）。他在1914年1月通过口试后以最优等成绩获得博士学位，他的论文于1915年发表在奥古斯都·利奥波德·克雷勒的期刊上。他决定要成为一名学者[5]：-

……尽管犹太人晋升的可能性有限……

并立即开始撰写他的教授资格论文（Habilitation）学位论文。

当然，1914年毕业立即因第一次世界大战的爆发而带来了问题。I. Fraenkel在德国军队服役，主要参与医疗队，但也有一小段时间在气象部门服务。然而，他已经完成了足够的资格论文工作Über einfache Erweiterungen zerlegbarer Ringe Ⓣ（关于可分离环的简单扩张），使他能够在1915年将其提交给马尔堡大学，但此时他正在军队服役，无法参加考试和发表就职演讲，而这些是完成Privatdocent任命所必需的。1916年1月，他在塞尔维亚随军服役时重病，获准休假以恢复健康。他借此机会回到马尔堡，在那里参加了所需的考试，发表了他的就职演讲，并成为了一名编外讲师。然而，他此时无法开始教学，因为他必须返回继续服兵役。在军队服役期间，他致力于撰写他的第一本关于集合论的书， Einleitung in die Mengenlehre Ⓣ（集合论导论）。

第一次世界大战结束后退伍，弗兰克尔于1918-19年冬季学期开始在马尔堡大学授课。大约就在这时，他遇到了正在学习德语的Wilhelmina Malka A Prins。他们于1919年3月28日结婚[28]：-

[弗兰克尔]认为他们的结合是理想的，因为除了宗教和犹太复国主义之外，他们没有共同的兴趣。他们完美地互补，最终共同育有四个孩子[两个儿子Benjamin和Aviezri，两个女儿Tirza和Rachel]。

在战后的德国，找到住处非常困难，弗兰克尔和他的妻子住在库尔特·亨泽尔马尔堡家中的一套公寓里。弗兰克尔已将他关于集合论的书 Einleitung in die Mengenlehre Ⓣ（集合论导论）提交给出版商Teubner，但被拒绝了。然而，Springer同意出版这部著作，它于1919年问世。George Adam Pfeiffer（1889-1943）在第一次世界大战期间曾在美国陆军服役，担任普林斯顿大学的气象学教官，他在一篇评论中写道[24]：-

作者打算给出无穷集合论的导论，任何有足够兴趣和耐心的人都能理解。没有设定其他先决条件。作者说，他在最近的战争中积累了这种表述的经验，当时他通过向战场上的战友解释Mengenlehre来打发许多沉闷的时光。各章标题包括：集合的概念；等价与无穷集合的概念；可数集合；连续统；基数的概念；基数的可比性；基数的运算；有序集合与序型；线性点集；良序集合、良序及其意义；逻辑悖论与集合的概念。主题的选择以及每个主题的处理程度都确定得很好。科学上的诚实并没有为了（表面上）便于读者吸收而被牺牲。……这本书对大学高年级数学课程以及对数学哲学有一般兴趣的人应该非常有用。它应该有助于向更广泛的圈子介绍数学中一个基础而有趣的分支的思想和方法。

1921年，弗兰克尔出版了Die neueren Ideen zur Grundlageng der Analysis und Mengenlehre Ⓣ（微积分与集合论基础中的较新思想），次年又出版了Über den Begriff "definit" und die Unabhängigkeit des Auswahlaxioms Ⓣ（论“确定”概念与选择公理的独立性）、Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre Ⓣ（论Cantor-恩斯特·策梅洛集合论的基础）和Axiomatische Begründung der transfiniten Kardinalzahlen Ⓣ（超限基数的公理化辩护）。其中第三篇是献给库尔特·亨泽尔六十岁生日的。这些论文中呈现的工作源于弗兰克尔关于恩斯特·策梅洛集合论公理独立性的研究。特别是，他给出了某些公理的更简单表述，并添加了一条新的“替换”公理。弗兰克尔于1922年在马尔堡晋升为教授。1928年，他离开马尔堡，在基尔克里斯蒂安-阿尔布雷希特大学担任了一年正教授。他在[5]中明确表示，在马尔堡或基尔都没有对反犹主义的抱怨，但当然，他意识到了这个问题[5]：-

就大学而言，20世纪20年代的反犹倾向与1918年前的情况相比发生了逆转——也就是说，在巴伐利亚，这种倾向比德国北部和西部要明显得多。犹太人被任命为正教授的情况仍然少见，不过法兰克福和汉堡这两所新兴的城市大学是例外。

弗兰克尔是一位狂热的犹太复国主义者，他于1929年离开基尔，前往耶路撒冷希伯来大学任教，此时距该校成立已有四年。人们或许会以为，作为一名犹太复国主义者，弗兰克尔会热衷于接受耶路撒冷希伯来大学的邀请，但这样想就过于简单化了。事实上，他非常担心，由于希伯来大学是一所世俗机构，它可能并不适合他。他此前曾写过一篇文章，在其中表达了担忧：

……希伯来大学会成为对《圣经》和犹太教神圣文本进行异端“科学”研究的论坛。

他写信给拉比亚伯拉罕 Isaac Kook，询问是否因为其世俗性质，他应该接受希伯来大学的邀请去教书。拉比回答说，他应该去那里工作，以提高其精神水平。弗兰克尔在抵达巴勒斯坦后，将他的名字从弗兰克尔改为亚伯拉罕。在希伯来大学，弗兰克尔成为学院院长。他与阿尔伯特·爱因斯坦进行了热烈的通信，后者对希伯来大学和成为校长的可能性深感兴趣。他写信给弗兰克尔说：-

……没有比你更客观、更纯粹的评判者了……

以向他提供准确的信息。阿尔伯特·爱因斯坦还就希伯来大学的物理学教职任命向弗兰克尔提供了建议。此时，希伯来大学在其未来方向上存在许多困难，弗兰克尔围绕这些问题进行了许多斗争。

他曾获准从基尔休假两年，但本希望能留在希伯来大学。然而在耶路撒冷两年后，那里糟糕的经济状况意味着他请求休假，并于1931年夏天回到基尔的职位。回到基尔后，Erhard Tornier 找到他，向弗兰克尔请求在基尔的一个职位，以便能向他学习。弗兰克尔同意了，但1932年，Tornier 加入了纳粹党，而在1933年初希特勒上台后，Tornier 明确表示他想继承弗兰克尔的讲席。1933年4月1日，发生了所谓的“抵制日”，犹太商店遭到抵制，犹太讲师不被允许进入大学。1933年4月7日，《公务员法》获得通过，该法提供了将犹太教师从大学中清除的手段，当然也清除了其他职位上的犹太裔人士。所有非雅利安血统的公务员（祖父母中有一人信奉犹太教即被视为非雅利安人）都应退休，但对于在第一次世界大战中为德国作战的人有豁免。I. Fraenkel符合这一豁免条款，但在4月25日，他申请休假。他收到了院长的回复，Wesle 教授，一位纳粹任命的人，声明[3]：-

……满意地指出，在基尔大学整个任职期间，他一直宣称自己是犹太人而非德国人。

他寻求回到耶路撒冷希伯来大学，并于9月9日获准休假。10月，他获准变更住所。起初他在耶路撒冷的日子很艰难，因为他在领取薪水方面遇到问题。即使他确实收到了部分应得的钱，也是付入一个德国账户，从巴勒斯坦几乎无法取用。

弗兰克尔将在希伯来大学度过余下的职业生涯，被任命为数学系首任院长，并在1938-40年间担任大学校长。正是在这些年间，亚伯拉罕·鲁滨逊——他于1933年还是高中生时从美国移民到巴勒斯坦——是他的学生。这对希伯来大学来说是困难时期，大学面临严重的财政困难。它无力继续支付弗兰克尔的薪水，但被纽约的美国希伯来大学之友所拯救，他们介入资助了弗兰克尔的讲席。

战争结束后，基尔大学于1946年3月找到他，请他回到那里的数学讲席。他拒绝了，并于1946年6月18日写信给基尔哲学学院院长 Unsöld 教授[3]：-

……现在期望我离开我的人民、我的国家和我的大学，去另一个国家担任讲席，这将是荒谬的。然而，即使不考虑我的个人态度……我认为，即使从纯粹客观的角度来看，任何犹太人要再次生活在一个其人口——大部分是主动地，其余几乎完全是被动地——曾对超过五百万犹太人（我族人的三分之一）在数千年未经历的残酷条件下遭到灭绝负有责任的国家，这也是一个不可能的想法。生活在这样一个民族中间将是无法忍受的。

1947年2月2日，弗兰克尔写信给 Erich Kamke（1890-1961），后者于1937年因其妻子是犹太人而被解除教授职务。弗兰克尔写道（用英语）[3]：-

我感到渴望表达我真诚的祝愿，向那些——唉，非常少的——尚未投身纳粹主义的前同事们，我想向您致以最亲切的个人问候。我猜想图宾根在战争期间遭受的损失比大城市要小，我满怀信心地希望您一切安好。最近在美国逗留的几个月里，我听说了相当多的同事，但没有得到您的消息。我想知道图宾根大学是否仍在运作，或者您是否已经去了别的地方。您会理解我不得不拒绝回到基尔的召唤。在一个对五百万犹太人的残酷谋杀负有责任的国家里，我无法呼吸。

弗兰克尔从1914年起一直是德国数学会的成员，直到1939年他的成员资格被终止。该学会由Erich Kamke于1948年重新建立，他写信给那些曾被驱逐的人，邀请他们重新加入。弗兰克尔拒绝了，后来在1951年1月，他再次写信给Kamke，确认收到重新启动的Jahresbericht及其纪念表，表中列出了自1933年以来去世的数学家，他说纪念表和新民主精神令他满意，但他仍然无法重新加入。

如上所述，弗兰克尔的第一项工作是研究库尔特·亨泽尔的$p$进数和环论，但他最著名的是在集合论方面的工作，于1919年出版了这一主题的第一部重要著作Einleitung in die Mengenlehre Ⓣ（集合论导论）。他在1922年和1925年两次尝试将集合论置于一种能避免悖论的公理框架中。他试图改进恩斯特·策梅洛的定义，并在他的公理系统内证明了选择公理的独立性。他的公理系统由陶拉尔夫·斯科伦在1922年和1928年修改，形成了今天所称的ZFS系统。该系统以恩斯特·策梅洛、弗兰克尔和陶拉尔夫·斯科伦命名。在这个系统中，证明选择公理的独立性更加困难，直到1963年科恩的工作才实现。弗兰克尔写了几部重要的著作。

你可以在THIS LINK看到这些书的一些评论摘录。

弗兰克尔还对数学史感兴趣，并撰写了多部关于该主题的重要著作。他于1920年在Materialien für eine wissenschaftliche Biographie von GaussⓉ（卡尔·弗里德里希·高斯科学传记资料）中撰写了关于卡尔·弗里德里希·高斯在代数方面的工作，然后于1930年在Jahresbericht der Deutschen Mathematiker-Vereinigung中出版了格奥尔格·康托尔的重要传记。同样在1930年，他出版了Beliefs and Opinions in Light of the Natural Sciences（希伯来文），其中他试图根据他的宗教观点来看待科学的进步。例如他写道（见[29]中的英文翻译）：-

……当《托拉》描述世界的形成时，更一般地说，当它谈论自然界中发生的事情时，原则上，它使用的规则是“《托拉》以人的语言说话”。这不仅是因为这条规则适合《托拉》的目的，即不是解释自然科学；而是它必须如此，这是至关重要的。如果《托拉》描述了自然科学的精确过程，那么在该科学水平被发现之前的时期，它就不会被理解。每一代人都必须根据他们科学理论的进展来改变他们对《托拉》的立场。

弗兰克尔还发表了许多关于犹太数学和犹太数学家的文章，例如Jewish mathematics and astronomy（1960年）。

弗兰克尔的许多学生对数学做出了重要贡献，包括亚伯拉罕·鲁滨逊，他在1957年从希伯来大学讲席退休后接替了他。弗兰克尔在希伯来大学的时光是充满困难的时期之一，尤其是在他职业生涯的最后十年。1948年进行的独立战争使耶路撒冷被约旦和以色列分割。位于斯科普斯山顶的希伯来大学处于非军事区，每两周只能通过车队进入一次。退休后，弗兰克尔没有放弃教学，继续在特拉维夫附近的巴伊兰大学讲课。

我们已经提到弗兰克尔是一位热忱的犹太复国主义者。因此，他参与了政治活动，在英国托管时期是巴勒斯坦犹太国民议会执行委员会Vaad Leumi的成员。弗兰克尔也是犹太复国主义内部Merkaz Ruhani宗教运动的成员。该党促进犹太宗教教育，建立宗教学校，并大力推动首席拉比在婚姻和离婚等所有犹太事务上的权威。

最后，让我们提到弗兰克尔的兄弟Joel Eugen与Esther结婚，他们有一个儿子Aviezri，他成为了一位数学家，对组合博弈论做出了重大贡献。

Abraham Fraenkel's parents were Sigmund Aviezri Fraenkel (1860-1925) and his wife Charlotte Chaja Sara Neuburger (1868-1965), both of whom were born in Munich, Bavaria. Sigmund, who was a wool merchant, and Charlotte were married in Munich on 3 November 1889. We should note at this point that the family was Jewish and in fact Sigmund, in addition to being a merchant, was a leader of Orthodox Judaism in Bavaria and an economist working for the German government. Abraham, the subject of this biography, was known as Adolf until later in his life when he went to Palestine and from that time on called himself Abraham. However, to avoid confusion, we will refer to him as Abraham throughout this biography. He was the eldest of his parents' children having younger siblings Joel Eugen (born 25 March 1892), Else Lea (born 11 July 1896), and Paula Tirza (born 29 May 1901).

Fraenkel began his education at the age of five when his parents employed a private tutor for the young child. He learnt Hebrew from this private tutor and then attended an elementary school in Munich. He entered the Luitpold Gymnasium which had been founded by Prince Luitpold of Bavaria in 1891, the year in which Fraenkel was born. This Gymnasium served the eastern part of Munich and was situated in the Alexandrastrasse opposite the National Museum. Fraenkel was an outstanding pupil, receiving the grade "very good" in every subject he studied. He graduated from the Gymnasium in July 1909. He had already written articles on the religious calendar systems: Bestimmung des Datums des jüdischen Osterfestes für die Zeitrechnung der Mohammedaner Ⓣ (1908); Eine Formel zur Verwandlung jüdischer Daten in mohammedanische Ⓣ (1909); and Die Berechnung des Osterfestes Ⓣ (1910). These articles combined his interests in religion and mathematics.

In common with most students in Germany in his time, Fraenkel studied for periods at different universities. He spent a semester at the Ludwig-Maximilian University of Munich, then went to the University of Marburg where he attended courses by Kurt Hensel, Ernst Richard Neumann (1875-1955) and Ernst Hellinger. He then went to the Friedrich-Wilhelm University of Berlin (which has been known as the Humboldt University of Berlin since 1949) where he was taught by Hermann Amandus Schwarz, Georg Frobenius and Friedrich Schottky. He spent the final year of his studies at the University of Breslau before returning to the University of Marburg where, having been advised by Kurt Hensel, he submitted his doctoral thesis Über die Teiler der Null und die Zerlegung von Ringen Ⓣ. He was awarded his doctorate summa cum laude after his oral examination in January 1914 and his thesis was published in Crelle's Journal in 1915. He decided that he wanted to become an academic [5]:-

... despite the restricted possibilities for advancement open to Jews ...

and immediately began working on his habilitation thesis.

Of course, graduating in 1914 immediately presented problems due to the outbreak of World War I. Fraenkel served in the German army, mostly involved in the medical corps but also for a short time in the meteorological service. However, he had completed enough work on his habilitation thesis Über einfache Erweiterungen zerlegbarer Ringe Ⓣ that he was able to submit it to the University of Marburg in 1915 but, by this time serving in the army, he was not in a position to take the examinations and deliver the inaugural lecture which were required to complete an appointment as a Privatdocent. In January 1916 he was serving with the army in Serbia when he was taken seriously ill and allowed leave in which he might recover his health. He took the opportunity to return to Marburg where he took the required examinations, gave his inaugural lecture and became a Privatdocent. However, he could not begin teaching at this time as he had to return to continue his army service. While he was serving in the army he worked on writing his first book on set theory, Einleitung in die Mengenlehre Ⓣ.

After being released from military service following the end of World War I, Fraenkel began lecturing at the University of Marburg in the winter semester of 1918-19. It was around this time that he met Wilhelmina Malka A Prins who was studying German. They married on 28 March 1919 [28]:-

[Fraenkel] thought their partnership was ideal because apart from religion and Zionism, they had no interests in common. They complemented each other perfectly and would eventually have four children together [two sons Benjamin and Aviezri, two daughters Tirza and Rachel].

Finding a home in which to live was very difficult in post-war Germany and Fraenkel and his wife lived in a flat in Kurt Hensel's Marburg home. Fraenkel had submitted his set theory book Einleitung in die Mengenlehre Ⓣ to the publisher Teubner, but it had been rejected. However, Springer agreed to publish the work and it appeared in 1919. George Adam Pfeiffer (1889-1943), who had served in the U.S. Army as an instructor in meteorology at Princeton University during World War I, writes in a review [24]:-

The author proposes to give an introduction to the theory of infinite sets which can be understood by anyone who has sufficient interest and patience. No other prerequisites are set down. The author tells that he had experience in this sort of presentation during the recent war, when he lightened many wearisome hours by explaining Mengenlehre to his comrades in the field. Among the chapter headings are: the concept of set; the concepts of equivalence and infinite sets; countable sets; the continuum; the concept of cardinal number; comparability of cardinal numbers; operations on cardinal numbers; ordered sets and types of order; linear point sets; well-ordered sets, well-ordering and its significance; logical paradoxes and the concept of set. The choice of topics and the extent to which each topic is treated are well determined. Scientific honesty is not sacrificed for (apparently) easy assimilation by the reader. ... The book should be very useful for upper collegiate classes in mathematics and for those interested in mathematical philosophy in a general way. It should help to introduce to a wider circle the ideas and methods of a fundamental and interesting branch of mathematics.

In 1921 Fraenkel published Die neueren Ideen zur Grundlageng der Analysis und Mengenlehre Ⓣ and, in the following year, Über den Begriff "definit" und die Unabhängigkeit des Auswahlaxioms Ⓣ, Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre Ⓣ and Axiomatische Begründung der transfiniten Kardinalzahlen Ⓣ. The third of these was dedicated to Kurt Hensel for his sixtieth birthday. This work presented in these papers came out of Fraenkel's research on the independence of Zermelo's set theory postulates. In particular he gave simpler formulations of certain axioms and added a new axiom of "replacement". Fraenkel was promoted to professor in Marburg in 1922. In 1928 he left Marburg and spent one year as a full professor at the Christian-Albrecht University of Kiel. He states clearly in [5] that had no complains about anti-Semitism at either Marburg or Kiel but, of course, he was aware of the problem [5]:-

As concerns the universities, anti-Semitic tendencies were in the 1920s reversed from the situation before 1918 - that is, in Bavaria they were much more pronounced than in the North and West of Germany. The naming, though not the preferment, of Jews to positions as full professors remained infrequent with the exception of the new city universities of Frankfurt and Hamburg.

Fraenkel was a fervent Zionist and he left Kiel to teach at the Hebrew University of Jerusalem from 1929, joining the university four years after its foundation. Now one might imagine that Fraenkel, as a Zionist, would be enthusiastic about accepting an invitation from the Hebrew University of Jerusalem but this would be an over simplification. In fact he was very concerned that, since the Hebrew University was a secular institution, it might not be an appropriate place for him. He had earlier written an article where he expressed concerns:-

... that the Hebrew University would be a forum for heretical "scientific" studies of the Bible and Jewish sacred texts.

He wrote to Rabbi Abraham Isaac Kook asking for advice on whether, because of its secular nature, he should accept an invitation to teach the Hebrew University. The Rabbi replied that he should go there and work to raise its spiritual level. Fraenkel changed his first name from Adolf to Abraham after his arrival in Palestine. At the Hebrew University, Fraenkel became dean of the faculty. He entered into a vigorous correspondence with Albert Einstein who was deeply interested in the Hebrew University and the possibility of becoming rector. He wrote to Fraenkel saying:-

... there is no more objective and pure judge than you ...

to give him accurate information. Einstein also gave Fraenkel advice on physics appointments to the Hebrew University. At this time there were many difficulties at the Hebrew University regarding its future direction, and Fraenkel had many battles over these issues.

He had been given a two year leave from Kiel but had hoped to be able to remain at the Hebrew University. After two years in Jerusalem, however, the poor economic situation there meant that he requested leave and, in the summer of 1931, returned to his position at Kiel. Back in Kiel, he was approached by Erhard Tornier who asked Fraenkel for a position in Kiel so that he might learn from him. Fraenkel agreed but, in 1932, Tornier joined the Nazi party and, after Hitler came to power in the early part of 1933, Tornier made it clear that he wanted to inherit Fraenkel's chair. On 1 April 1933 there was the so-called "boycott day" when Jewish shops were boycotted and Jewish lecturers were not allowed to enter the university. On 7 April 1933 the Civil Service Law was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired but there were exemptions for those who had fought for Germany in World War I. Fraenkel qualified under this exemption clause but, on 25 April, he applied for leave of absence. He received a reply from the Dean of the Faculty, Professor Wesle, who was a Nazi appointment, stating [3]:-

... with satisfaction that during his whole service at Kiel University, he had always declared himself to be a Jew and not a German.

He sought to return to the Hebrew University of Jerusalem and, on 9 September his leave was granted. In October he was given permission to change his residence. Initially his time in Jerusalem was difficult since he had problems getting his salary. Even when he did receive some of the money that he was due, it was paid into a German account which was almost impossible to access from Palestine.

Fraenkel was to spend the rest of his career at the Hebrew University, being appointed the first Dean of the Faculty of Mathematics, and serving as the rector of the university during 1938-40. It was during these years that Abraham Robinson, who had emigrated from the United States to Palestine when a high school pupil in 1933, was his student. These were difficult times for the Hebrew University which had severe financial difficulties. It could not afford to continue to pay Fraenkel's salary but it was saved by the American Friends of the Hebrew University in New York who stepped in to subsidise Fraenkel's chair.

After the war ended, he was approached by the University of Kiel in March 1946 to return to the chair of mathematics there. He refused, writing to the Dean of the Philosophical Faculty of Kiel, Professor Unsöld, on 18 June 1946 [3]:-

... it would be absurd to expect me now to leave my people, my country and my University for a chair in another country. Even without, however, taking into account my personal attitude ... I think it would be even from a purely objective point of view, an impossible idea for any Jew to live again in a country whose population - to a large extent actively and for the rest almost entirely passively - has been responsible for the extermination of more than five millions of Jews, the third part of my People, under conditions of cruelty not experienced for thousands of years. It would be intolerable to live among such a nation.

On 2 February 1947, Fraenkel wrote to Erich Kamke (1890-1961) who had been dismissed from his professorship in 1937 because his wife was Jewish. Fraenkel wrote (in English) [3]:-

Feeling the desire of expressing my sincere wishes to the, alas, very few of my former colleagues who have not delivered themselves to Nazism, I should like to send you my kindest personal regards. I assume that Tübingen has suffered less than the big cities during the war, and I confidently hope that you are well. On a recent stay in U.S.A. during a few months, I heard of quite a number of colleagues, but I did not obtain news about you. I wonder if the Tübingen University still works, or if you have gone over to another place. You will understand that I had to decline the call to go back to Kiel. In a country being responsible for the cruel murder of five million Jews I could not breath.

Fraenkel had been a member of the German Mathematical Society from 1914 until his membership was terminated in 1939. The Society was re-established by Erich Kamke in 1948 and he wrote to those who had been expelled inviting them to rejoin. Fraenkel refused and, later in January 1951, he again wrote to Kamke acknowledging receipt of the restarted Jahresbericht with its memorial table of those mathematicians who had died since 1933, saying that the memorial table and the new democratic spirit pleased him, but he still could not rejoin.

As we saw above, Fraenkel's first work was on Hensel's $p$-adic numbers and on the theory of rings, but he is best known for his work on set theory, publishing his first major work on the topic Einleitung in die Mengenlehre Ⓣ in 1919. He made two attempts, in 1922 and 1925, to put set theory into an axiomatic setting that would avoid the paradoxes. He tried to improve the definitions of Zermelo and, within his axiom system, he proved the independence of the axiom of choice. His system of axioms was modified by Skolem in 1922 and 1928 to give what is today known as the ZFS system. This is named after Zermelo, Fraenkel and Skolem. Within this system it is harder to prove the independence of the axiom of choice and this was not achieved until the work of Paul Cohen in 1963. Fraenkel wrote several important books.

You can see some brief extracts from reviews of these books at THIS LINK

Fraenkel was also interested in the history of mathematics and wrote a number of important works on the topic. He wrote on Carl Friedrich Gauss's work in algebra in 1920 in Materialien für eine wissenschaftliche Biographie von Gauss Ⓣ, then in 1930, he published an important biography of Georg Cantor in the Jahresbericht der Deutschen Mathematiker-Vereinigung. Also in 1930 he published Beliefs and Opinions in Light of the Natural Sciences (Hebrew) in which he attempted to look at the progress of science in the light of his religious views. For example he writes (see the English translation in [29]):-

... when the Torah describes the formation of the world, and more generally when it speaks about what goes on in nature, in principle, it uses the rule that "the Torah speaks in the language of people." It is not only that this rule is suitable with respect to the purpose of the Torah, which is not to explain the natural sciences; rather it is essential that it be this way. Had the Torah described the precise processes of natural science, then it would not have been understood in the periods prior to the discovery of that level of science. Each generation would have had to change its stance vis-à-vis the Torah in accordance with the progress of their scientific theories.

Fraenkel also published many articles relating to Jewish mathematics and Jewish mathematicians, for example Jewish mathematics and astronomy (1960).

A number of Fraenkel's students have made important contributions to mathematics including Abraham Robinson who succeeded him when he retired from his chair at the Hebrew University in 1957. Fraenkel's time at the Hebrew University had been one of many difficulties, particularly in the last ten years of his career. The War of Independence fought in 1948 left Jerusalem divided between Jordan and Israel. The Hebrew University, on the top of Mount Scopus, was in a demilitarised zone and was only accessible by convoy once every two weeks. After retiring Fraenkel did not give up teaching, continuing to lecture at Bar Ilan University, near Tel Aviv.

We have already mentioned that Fraenkel was a fervent Zionist. As such he was involved in political activity being a member of the Vaad Leumi, the executive committee of the Palestinian Jewish National Assembly at the time of British mandate. Fraenkel was also a member of the Merkaz Ruhani religious movement within Zionism. This party promoted Jewish religious education, established religious schools and strongly promoted the authority of the chief rabbinate over all Jewish matters such as marriage and divorce.

Finally, let us mention that Fraenkel's brother Joel Eugen married Esther and they had a son Aviezri who became a mathematician making major contributions to combinatorial game theory.