阿尔弗雷德·塔斯基Alfred Tarski

阿尔弗雷德·塔斯基的父亲是Ignacy Teitelbaum，一位犹太店主和商人，从事木材贸易。人们可能会合理地问为什么塔斯基的父亲不叫“塔斯基”，我们稍后会解释为什么塔斯基 Teitelbaum改名为塔斯基。现在只需注意，塔斯基实际上出生时姓Teitelbaum，在他生命的前22年左右被称为塔斯基 Teitelbaum。Ignacy Teitelbaum娶了Rosa Prussak，尽管Rosa从未有过职业，因此从未有机会展示她的才智，但塔斯基的才华是通过他的母亲而不是父亲继承的。塔斯基有一个兄弟Wacław Teitelbaum，他们在成长过程中关系亲密。

塔斯基在一个重视教育的富裕家庭长大。他就读于Schola Mazowiecka（Nizina Mazowiecka是波兰低地地区的一部分，华沙就位于其中），这是一所为知识分子设立的高中。这所学校给了塔斯基比否则更广泛的教育。除了标准学校科目外，他还学习了俄语、德语、法语、希腊语和拉丁语等科目。当然，他在高中学习了数学，他的老师认识到他在该科目上有非凡的才能，但这并不是他打算在大学专攻的科目，而是决定学习生物学。

在塔斯基 Teitelbaum成长的岁月里，波兰发生了重大变化，我们需要简要回顾一下背景，以便理解所发生的事件。自1772年波兰被瓜分以来，俄罗斯控制了该国称为波兰会议王国的那部分，其中包括华沙。华沙大学已被关闭，只有一所俄语大学在那里运作。第一次世界大战爆发时，同盟国（德国和奥匈帝国）进攻了波兰会议王国。1915年8月，俄军从华沙撤退——此时塔斯基 Teitelbaum正在读高中，我们必须将他辉煌的学业生涯置于他周围发生的军事行动和剧烈政治变革的背景下来看。德国和奥匈帝国控制了该国大部分地区，并在华沙设立了一位德国总督。

俄国撤退后的首批举措之一就是重建华沙大学，它于1915年11月开始作为一所波兰大学运作。一所强大的数学学派迅速在该大学成长起来。扬·武卡谢维奇在1915年华沙大学重新开放时被任命到该校。斯特凡·马祖尔凯维奇此时成为数学教授。这是波兰激动人心的时期，1916年11月5日宣布成立了新的波兰王国。塔斯基 Teitelbaum（他仍未将名字改为塔斯基）离开学校后在波兰军队中待了很短一段时间，然后于1918年进入华沙大学，开始了一门他打算最终获得生物学学位的课程。这是一个极其激动人心的时期，因为这所大学几乎在一夜之间成为了一所领先的国际机构。

斯坦尼斯瓦夫·雷斯涅夫斯基于1919年在华沙接受了数学哲学讲席，瓦茨瓦夫·谢尔宾斯基同时被任命。这些任命意义重大，因为塔斯基选修了斯坦尼斯瓦夫·雷斯涅夫斯基讲授的逻辑课程，后者很快看出了他的天才，并说服他从生物学转向数学。这对塔斯基来说是一个决定性时刻，他现在不仅受到斯坦尼斯瓦夫·雷斯涅夫斯基的影响，还受到扬·武卡谢维奇、瓦茨瓦夫·谢尔宾斯基、斯特凡·马祖尔凯维奇以及哲学家Kotarbinski的影响。他参加了所有这些领先学者的课程，并迅速展现出与他的老师们同样的才华，从而显示了他的天才。1920年，他在学习中途曾在波兰军队中短暂服役。

大约在1923年，塔斯基 Teitelbaum将名字改为塔斯基。改名以及他决定同时做出的另一项重大生活改变——即从犹太教改信罗马天主教——有许多原因。做出这两项重大改变的不仅仅是塔斯基，因为他的兄弟Wacław同时改名为塔斯基并皈依了罗马天主教。我们已经解释了波兰在此前几年所经历的新开端，这个国家有着强烈的民族主义情绪。毫无疑问，塔斯基深受这些情绪的影响，希望成为一个波兰人而不是犹太人。改名和改宗都使他更加波兰化。他还意识到，这个国家的反犹观点使犹太人几乎不可能被任命到大学职位，而塔斯基在博士学业接近尾声时，当然希望从事学术生涯。

塔斯基的第一篇论文发表于1921年，当时他年仅19岁。在这篇论文中，他研究了集合论问题，事实上集合论将成为塔斯基一生持续的研究兴趣。他的博士研究由斯坦尼斯瓦夫·雷斯涅夫斯基指导，并于1923年提交博士论文送审。1924年，塔斯基获得博士学位，成为华沙大学有史以来最年轻的博士学位获得者。塔斯基的第一批重要成果发表于1924年，当时他开始在格奥尔格·康托尔、恩斯特·策梅洛和理查德·戴德金获得的集合论结果基础上进行研究。同年，他与斯特凡·巴拿赫发表了一篇联合论文，研究现在所谓的斯特凡·巴拿赫-塔斯基悖论。这根本不是悖论，之所以得此名，唯一原因是它反直觉。该结果证明，一个球体可以被切成有限数量的碎片，然后重新组装成一个更大的球体，或者可以重新组装成两个与原球体大小相等的球体。

塔斯基于1922年至1925年在华沙的波兰教育学院教授逻辑学，同年被任命为华沙大学的数学与逻辑学讲师（Docent）。他后来成为扬·武卡谢维奇的助手，但这些大学职位不足以维持生计，因此他不得不靠第二份工作谋生。1925年，他成为华沙Zeromski中学的数学教授，并一直担任这两个全职职位直到1939年。1929年6月23日，塔斯基与Maria Witkowski结婚，后者是Zeromski中学的教师。鉴于我们上文提到的塔斯基的波兰爱国主义，值得注意的是Maria是罗马天主教徒，并曾在波兰争取独立的斗争中担任军队的信使。

这是塔斯基的国际声誉持续增长的时期。他于1930年2月访问了维也纳大学，并在卡尔·门格尔的讨论班上讲课。在维也纳，他遇到了库尔特·弗雷德里希·哥德尔，后者最近获得了博士学位，并于当年晚些时候成为教员。1933年，塔斯基发表了The concept of truth in formalized languages，这是他现在著名的关于真理概念的论文：-

……这是关于数理逻辑的最重要论文之一。……这篇论文不仅对早期数理逻辑中发展起来的若干思想给出了数学上严格的表述，还为后来的逻辑奠定了基础。

塔斯基获得了一笔奖学金，使他得以于1935年1月返回维也纳，并与卡尔·门格尔的研究小组一起工作到6月。逻辑实证主义的维也纳学派，特别是在20世纪20年代的维也纳蓬勃发展，促成了科学统一小组的发展，该小组于1935年在巴黎开会。塔斯基在这次会议上的一次演讲中提出了他关于真理的思想。Niiniluoto在[47]中论证道：-

……为这样一个论点：塔斯基关于真理的原始定义，连同其在模型论中的后期阐发，是对经典符合论真理观的一种阐释。在为塔斯基辩护以回应其某些批评者时，我希望表明这种真理观如何能够以哲学上令人满意的方式被表述、理解并进一步发展。

塔斯基于1936年发表了On the concept of logical consequence。在其中他主张，一个论证的结论逻辑地从其前提推出，当且仅当前提的每一个模型都是结论的模型。这项关于逻辑后承的工作产生了深远影响，并被许多作者讨论；例如参见[5]、[13]、[23]、[36]、[53]、[56]和[70]。

1937年，他发表了另一篇经典论文，这次是关于演绎方法的，其中清楚地阐述了他对演绎方法的性质和目的的看法，并考虑了逻辑在科学研究中的作用。

1939年，塔斯基申请了Lwów的哲学讲席，但未能获得任命。很难确定为什么在讲席竞争中一位候选人可能比另一位更受青睐，因此无法肯定地说反犹太主义在决定中起了作用。然而，可以肯定地说，到1939年，塔斯基已拥有杰出的国际声誉，却仍然被迫通过在中学教授数学来维持生计。同样有理由相信，改换姓名和宗教并不能让犹太人逃脱当时在整个欧洲普遍存在的歧视。

1939年8月，塔斯基前往美国哈佛大学参加另一次科学统一会议。1939年8月31日12时40分，希特勒下令他的部队于次日凌晨4时45分进攻波兰。当时塔斯基已在美国待了两周。对他来说极其幸运的是，德国军队进攻时他不在波兰，因为毫无疑问，尽管改了姓名和宗教，就纳粹政权而言，他仍然会被认定为犹太人。

此时塔斯基已有两个孩子，一个儿子Jan和一个女儿Ina，当他1939年前往美国时，他的妻子和两个孩子都留在波兰。塔斯基成功获得了留在美国的许可，随后他试图在许多欧洲朋友的帮助下安排家人逃离并到美国与他团聚。他未能实现这一目标，但幸运的是，三人都在战争中幸存下来，并于1946年与塔斯基团聚。然而，他的父亲、母亲、兄弟和嫂子都在战争期间死于纳粹之手。

塔斯基的生命因身处美国而得以保全，但他仍须谋得一份工作。永久职位并不容易获得，因为许多杰出学者在战争爆发前几年已从欧洲逃往美国。塔斯基担任过若干临时研究职位：1939至1941年在哈佛；1940年在纽约市立学院；1941至42年在普林斯顿的爱德华·斯图迪高等研究院，当时他持有古根海姆奖学金。在普林斯顿，塔斯基再次遇到了库尔特·弗雷德里希·哥德尔，因为后者同样逃离了纳粹的威胁。在此期间，1941年，他发表了一篇关于关系演算的重要论文。

在经历了这些年的临时职位之后，塔斯基于1942年加入位于乔治·伯克利的加利福尼亚大学教职员队伍，从而获得了一个永久职位。起初他被任命为一个为期一年的职位，但很快便获得了终身教职。他于1945年被提升为那里的副教授，1949年成为数学教授。他在伯克利度过了余下的职业生涯，于1968年成为荣休教授。尽管那时已正式退休，他仍被请求继续授课直到1973年，并且还继续指导研究生和从事研究，直至去世。在[3]中，我们了解到塔斯基在伯克利的学生情况。

他在伯克利的讨论班迅速成为逻辑学的重镇。他的学生——其中许多人如今已是著名数学家——回忆起他以令人敬畏的精力哄劝和诱导他们做出最好的工作，始终要求最高标准的清晰性和精确性。

塔斯基在伯克利当然没有过一种平静的生活，而是抓住许多机会访问其他地方。他于1950年担任伦敦大学学院的Sherman纪念讲师，随后于1955年担任巴黎儒勒·昂利·庞加莱研究所的讲师。他于1958—1960年担任米勒基础科学研究所的研究教授，之后于1966年回到伦敦大学学院，再次担任Sherman纪念讲师。1967年，他担任洛杉矶加利福尼亚大学的Flint哲学教授，并于1974—75年在南美洲的智利天主教大学。

塔斯基被公认为有史以来最伟大的四位逻辑学家之一，另外三位是亚里士多德、戈特洛布·弗雷格和库尔特·弗雷德里希·哥德尔。其中塔斯基作为逻辑学家最多产，他的文集不包括著作就有2500页。塔斯基在数学的许多领域做出了重要贡献：集合论、测度论、拓扑学、几何学、经典代数与泛代数、代数逻辑、形式逻辑和元数学的各个分支。他提出了‘逻辑后承’的公理，研究演绎系统、逻辑代数和可定义性理论。他可以被视为一位具有异常广泛数学兴趣的数学逻辑学家。我们已简要考察了塔斯基的一些工作，还将再审视一点他的工作，但在这样长度的传记中不可能恰当展现他贡献的范围。

元数学由大卫·希尔伯特于1922年引入，意为“证明论”，是其建立算术一致性纲领的一部分；塔斯基引入了语义方法，将其改造，从而发展出结合语义关系与句法关系的模型论。Sinaceur写道[60]：-

在塔斯基看来，元数学变得与任何数学学科相似。不仅其概念和结果可以被数学化，而且它们实际上可以被整合到数学中。……塔斯基摧毁了元数学与数学之间的界限。他反对将元数学的作用限制在数学基础方面。

形式科学语言可以通过他发展的语义方法进行更彻底的研究。他研究模型论、数学判定问题以及泛代数。

塔斯基于1957年12月26日至1958年1月4日在加利福尼亚大学伯克利分校举行的国际研讨会上提交了他的论文The axiomatic method : with special reference to geometry and physics。他的论文发表于1959年的会议论文集，给出了他的几何公理系统。这一公理系统为大卫·希尔伯特在Grundlagen der GeometrieⓉ（几何基础）中处理的问题提供了一种更新的方法。

1968年，塔斯基写了另一篇著名论文Equational logic and equational theories of algebras，其中他概述了当时等式逻辑的元数学，并给出了一些新结果和一些未解决的问题。次年，塔斯基的论文Truth and proof发表，其中：-

……是整个数理逻辑中最优秀的阐述性作品之一。

该论文考虑了库尔特·弗雷德里希·哥德尔的不完备性定理以及塔斯基的不可定义性定理，并审视了它们对数学中公理方法的影响。

塔斯基在数学的不同领域写了十九部专著。他的工作包括Geometry（1935）、Introduction to Logic and to the Methodology of Deductive Sciences（1936）、A decision method for elementary algebra and geometry（1948）、Cardinal Algebras（1949）、Undecidable theories（1953）、Logic, semantics, metamathematics（1956）和Ordinal algebras（1956）。

Introduction to Logic and to the Methodology of Deductive Sciences是一篇按逻辑学和公理学本科课程水平撰写的导论。在A decision method for elementary algebra and geometry中，塔斯基证明了实数的加法与乘法的一阶理论是可判定的，这与库尔特·弗雷德里希·哥德尔和丘奇的结果形成对比——后者证明了自然数的加法与乘法的一阶理论是不可判定的，这种对比对非专家来说确实令人惊讶。Cardinal Algebras呈现了对满足某些刻画基数算术性质的代数研究。在Undecidable theories中，塔斯基证明了group theory、格、抽象射影几何、闭包代数以及其他数学系统是不可判定的。他已经明确表示，一般判定问题无解，数学家们应当多么高兴。在1939-40年间于哈佛大学的一次演讲中，塔斯基说：-

……判定问题在其最一般形式下的解是否定的。……［想必］许多数学家在听到这一结果时都感到一种深深的宽慰。也许有时在失眠的夜里，他们会惊恐地想到那一刻：某个邪恶的元数学家会找到肯定的解，并设计出一台机器，使我们能够以纯粹机械的方式解决任何数学问题。……危险现在已经过去……［并且］数学家们……可以安心睡觉了。

在Ordinal algebras中，塔斯基定义了一个代数，它刻画了序型的加法理论的性质。它与他在Cardinal algebras中提出的代数最显著的不同在于其加法是非交换的。

塔斯基的论文集共四卷，由Steven R Givant和Ralph N McKenzie编辑。弗瑞兹·约翰Corcoran在评论这些卷时写道：-

数学界应当感谢Givant和McKenzie为编纂这部塔斯基著作的珍贵文集所付出的努力。只有当我们看到塔斯基的论文汇集在一处时，我们才能开始体会他对现代数学思想、特别是对现代数理逻辑的影响之广度和深度。没有塔斯基的贡献，我们今天所知道的数理逻辑几乎是不可想象的。

G H Moore在[1]中写道，描述了塔斯基的性格：-

塔斯基性格外向、机智敏捷、意志坚强、精力充沛、言辞犀利。他更喜欢合作研究——有时与同事通宵工作——并且对优先权非常讲究。

A B Feferman 也在 [25] 中描述了他性格的某些方面：-

作为一位魅力非凡的领袖和教师，以他精妙准确而又引人入胜的阐述风格著称，塔斯基 对学生有着令人望而生畏的高标准，但同时又非常鼓励人，尤其是对女性——这与普遍趋势相反。一些学生被吓跑了，但一圈弟子留了下来，其中许多人成为该领域世界知名的领袖。

塔斯基 因当选为 国家科学院、Royal Netherlands Academy of Sciences 和英国科学院院士而获得荣誉。他被任命为 Algebra Universalis 的名誉编辑，并于1944年至1946年担任符号逻辑协会主席，1956-57年担任国际科学史与科学哲学联合会主席。他获得了智利天主教大学（1975年）和马赛大学（1977年）的名誉学位。1981年，伯克利 授予他 伯克利 奖状。

Alfred Tarski's father was Ignacy Teitelbaum, a Jewish shopkeeper and businessman who traded in wood. One might reasonably ask why Tarski's father was not named "Tarski" and we will explain in a moment why Alfred Teitelbaum changed his name to Alfred Tarski. Let us just note for now that Alfred Tarski was actually born with the family name of Teitelbaum, and for about the first 22 years of his life was known as Alfred Teitelbaum. Ignacy Teitelbaum had married Rosa Prussak and, although Rosa never had a career, and therefore never had the opportunity to show her intellect, it was through his mother rather than his father that Tarski inherited his brilliance. Alfred had a brother Wacław Teitelbaum to whom he was close as they were growing up.

Alfred grew up in a well off family who valued education. He attended the Schola Mazowiecka (the Nizina Mazowiecka is the part of the lowland area of Poland in which Warsaw is situated) which was a High School for the intellectuals. The school gave Alfred a broader education than he would otherwise gave received. He studied subjects such as Russian, German, French, Greek and Latin in addition to the standard school topics. Of course he studied mathematics at the high school, and his teachers recognised that he had extraordinary talents in that topic, but it was not the subject that he intended to specialise in at university, rather he made the decision to study biology.

There were major changes in Poland during the years that Alfred Teitelbaum was growing up and we need to look briefly at the background in order to understand events. Since the partition of Poland in 1772, Russia had controlled that part of the country called Congress Poland, which included Warsaw. The University of Warsaw had been closed and only a Russian language university operated there. At the outbreak of World War I, the Central Powers (Germany and Austria- Hungary) attacked Congress Poland. In August 1915 the Russian forces withdrew from Warsaw - at this time Alfred Teitelbaum was studying in high school and we have to see his brilliant school career against the military action and dramatic political change taking place around him. Germany and Austria-Hungary took control of most of the country and a German governor general was installed in Warsaw.

One of the first moves after the Russian withdrawal was the refounding of the University of Warsaw and it began operating as a Polish university in November 1915. Rapidly a strong school of mathematics grew up in the University. Łukasiewicz was appointed to the new University of Warsaw when it reopened in 1915. Mazurkiewicz became a professor of mathematics at this time. It was an exciting time in Poland and a new Kingdom of Poland was declared on 5 November 1916. Alfred Teitelbaum (who had still not changed his name to Tarski) spent a short while in the Polish army after leaving school and then entered the University of Warsaw in 1918, beginning a course which he intended would lead to a degree in biology. It was a time of great excitement as the university had become a leading international institution almost overnight.

Lesniewski accepted the chair of the philosophy of mathematics at Warsaw in 1919 and Sierpiński was appointed at the same time. These appointments were of great significance since Alfred took a course on logic given by Lesniewski who quickly saw his genius and persuaded him to change from biology to mathematics. It was a defining moment for Alfred who now came under the influence not only of Lesniewski but also of Łukasiewicz, Sierpiński, Mazurkiewicz, and the philosopher Kotarbinski. He attended courses by all these leading academics and showed his genius by quickly matching the brilliance of his teachers. In 1920 he had a short spell in the Polish army in the middle of his studies.

It was around 1923 that Alfred Teitelbaum changed his name to Alfred Tarski. There were a number of reasons for the name change and for the other major change to his life which he decided to make at the same time, namely to change his religion from the Jewish faith to become a Roman Catholic. It was not just Alfred who made these two major moves, for his brother Wacław took the name of Tarski and embraced Roman Catholicism at the same time. We have explained the new beginnings that Poland had experienced in the few years before this, and there were strong nationalist feelings in the country. There is no doubt that Tarski was strongly influenced by these feelings and wished to be a Pole and not a Jew. Both the name change and the change of religion made him more Polish. There was also the realisation that anti-Semitic views in the country made it almost impossible for a Jew to be appointed to a university post and Tarski, nearing the end of his doctoral studies, certainly wished to follow an academic career.

Tarski's first paper was published in 1921 when he was only 19 years old. In this paper he investigated set theory questions, and in fact set theory would be a continuing research interest for Tarski throughout his life. His doctoral studies were supervised by Lesniewski and he submitted his doctoral thesis for examination in 1923. In 1924 Tarski graduated with a doctorate, and became the youngest person ever to be awarded the degree by the University of Warsaw. Tarski's first major results were published in 1924 when he began building on the set theory results obtained by Cantor, Zermelo and Dedekind. He published a joint paper with Banach in that year on what is now called the Banach-Tarski paradox. This is not a paradox at all, the only reason that it is given this name is that it is counter-intuitive. The result proves that a sphere can be cut into a finite number of pieces and then reassembled into a sphere of larger size, or alternatively it can be reassembled into two spheres of equal size to the original one.

Tarski taught logic at the Polish Pedagogical Institute in Warsaw from 1922 to 1925 then in that year he was appointed Docent in mathematics and logic at the University of Warsaw. He later became Łukasiewicz's assistant but these university positions did not give him enough money to live on, so he had to earn his living with a second job. He became a mathematics professor at Zeromski's Lycée in Warsaw in 1925 and until 1939 he held these two full-time posts. On 23 June 1929 Tarski married Maria Witkowski who was a teacher at Zeromski's Lycée. Given Tarski's Polish patriotism which we mentioned above, it may be relevant to note that Maria was a Roman Catholic and she had worked as a courier for the army during Poland's fight for independence.

This was a time when Tarski's international reputation continued to grow. He visited the University of Vienna in February 1930 where he lectured to Menger's colloquium. In Vienna he met Gödel who had recently been awarded his doctorate and who became a member of the faculty later in that year. In 1933 Tarski published The concept of truth in formalized languages which is his now famous paper on the concept of truth:-

... which is among the most important papers ever written on mathematical logic. ... Not only does this paper provide a mathematically rigorous articulation of several ideas that had been developing in earlier mathematical logic, it also presents foundations on which later logic could be built.

Tarski was awarded a fellowship to allow him to return to Vienna in January 1935 and he worked with Menger's research group until June. The Vienna Circle of Logical Positivists which flourished, particularly in the 1920s in Vienna, had led to the development of the Unity of Science group and this group met in Paris in 1935. Tarski presented his ideas on truth in a lecture at this meeting. Niiniluoto, in [47] argues:-

... for the thesis that Alfred Tarski's original definition of truth, together with its later elaboration in model theory, is an explication of the classical correspondence theory of truth. In defending Tarski against some of his critics, I wish to show how this account of truth can be formulated, understood, and further developed in a philosophically satisfactory way.

Tarski published On the concept of logical consequence in 1936. In this he claimed that the conclusion of an argument will follow logically from its premises if and only if every model of the premises be a model of the conclusion. This work on logical consequence has had a profound influence and has been discussed by many authors; see for example [5], [13], [23], [36], [53], [56], and [70].

In 1937 he published another classic paper, this time on the deductive method, which presents clearly his views on the nature and purpose of the deductive method, as well as considering the role of logic in scientific studies.

In 1939 Tarski applied for the chair of philosophy at Lwów but failed to be appointed. It is hard to be certain why one candidate may be preferred to another in a competition for a chair so it is impossible to say with certainty that anti-Semitism played a role in the decision. However, it is certainly fair to say that by 1939 Tarski had an outstanding international reputation but was still forced to support himself by teaching mathematics in a high school. It is certainly reasonable to believe that changing one's name and religion would not allow Jews to escape from the discrimination which was widespread throughout Europe at this time.

In August 1939 Tarski travelled to Harvard University in the United States to attend another Unity of Science meeting. At 12.40 on 31 August 1939 Hitler gave the order for his troops to attack Poland at 4.45 the next morning. Tarski had been in the United States for two weeks at the time. It was extremely fortunate for him that he was not in Poland when the German armies attacked, for there is no doubt that despite the change of name and religion, he would still have qualified as a Jew as far at the Nazi regime were concerned.

By this time Tarski had two children, a son Jan and a daughter Ina, and both his wife and children had remained in Poland when he travelled to the United States in 1939. Tarski was successful in obtaining permission to remain in the United States, and he then tried, with the help of many European friends, to arrange for his family to escape and join him in the United States. He failed to achieve this but fortunately all three survived the war and were able to join Tarski in 1946. However, his father, mother, brother and sister-in-law all died at the hands of the Nazis during the war.

Certainly Tarski's life was saved by being in the United States but he still had to secure a job. Permanent posts were not easy to obtain since many outstanding academics had fled from Europe to the United States in the years immediately prior to the outbreak of war. Tarski held a number of temporary research positions: Harvard from 1939 to 1941; City College of New York in 1940; and the Institute for Advanced Study at Princeton in 1941-42 when he held a Guggenheim Fellowship. At Princeton Tarski met Gödel again for he had also fled from the Nazi threat. During this period, in 1941, he published an important paper calculus of relations.

After these years of temporary jobs, Tarski obtained a permanent post after he joined the staff at the University of California at Berkeley in 1942. At first he was appointed to a one year post but he was soon given tenure. He was promoted to associate professor there in 1945, becoming Professor of Mathematics in 1949. He remained at Berkeley for the rest of his career, becoming professor emeritus in 1968. Although officially retired at that stage, he was asked to continue to teach until 1973 and he also continued supervising research students and undertaking research up to the time of his death. In [3] we are told about Tarski's students in Berkeley.

His seminars at Berkeley fast became a power-house of logic. His students, many of them now distinguished mathematicians, recall the awesome energy with which he would coax and cajole their best work out of them, always demanding the highest standards of clarity and precision.

Tarski certainly did not lead a quite life in Berkeley, but rather took many opportunities to visit other places. He was Sherman memorial lecturer at University College London in 1950, then a lecturer at the Henri Poincaré Institute in Paris in 1955. He was research professor at the Miller Institute of Basic Research in Science in 1958-1960, then in 1966 returned to University College London when he was again Sherman memorial lecturer. In 1967 he was Flint professor of philosophy at the University of California at Los Angeles and in 1974-75 he was in South America at the Catholic University of Chile.

Tarski is recognised as one of the four greatest logicians of all time, the other three being Aristotle, Frege, and Gödel. Of these Tarski was the most prolific as a logician and his collected works, excluding his books, runs to 2500 pages. Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics. He produced axioms for 'logical consequence', worked on deductive systems, the algebra of logic and the theory of definability. He can be considered a mathematical logician with exceptionally broad mathematical interests. We have looked briefly at some of Tarski's work and we shall examine a little more of his work but it is impossible in a biography of this length to give a proper view of the range of his contributions.

Metamathematics, introduced by Hilbert in 1922 meaning "proof theory" as a part of his programme to establish the consistency of arithmetic, was transformed by Tarski when he introduced semantic methods leading to his development of model theory with its combination of semantic and syntactic relations. Sinaceur writes [60]:-

In [Tarski's] view, metamathematics became similar to any mathematical discipline. Not only its concepts and results can be mathematized, but they actually can be integrated into mathematics. ... Tarski destroyed the borderline between metamathematics and mathematics. He objected to restricting the role of metamathematics to the foundations of mathematics.

Formal scientific languages can be subjected to more thorough study by the semantic method that he developed. He worked on model theory, mathematical decision problems and with universal algebra.

Tarski presented his paper The axiomatic method : with special reference to geometry and physics to the International Symposium held at the University of California at Berkeley from 26 December 1957 to 4 January 1958. His paper, appearing in the Proceedings of the Conference in 1959, gives his axiom system for geometry. This axiomatic system gives a newer approach to the problem which had been tackled by Hilbert in Grundlagen der Geometrie Ⓣ.

In 1968 Tarski wrote another famous paper Equational logic and equational theories of algebras in which he presented a survey of the metamathematics of equational logic as it then existed as well as giving some new results and some open problems. In the following year Tarski's paper Truth and proof appeared which:-

... is one of the finest pieces of expository writing in all of mathematical logic.

The paper considers Gödel's incompleteness theorem as well as Tarski's undefinability theorem and look at their consequences for the axiomatic method in mathematics.

Tarski wrote nineteen monographs in different areas of mathematics. His work includes Geometry (1935), Introduction to Logic and to the Methodology of Deductive Sciences (1936), A decision method for elementary algebra and geometry (1948), Cardinal Algebras (1949), Undecidable theories (1953), Logic, semantics, metamathematics (1956), and Ordinal algebras (1956).

Introduction to Logic and to the Methodology of Deductive Sciences is an introduction written at the level of an undergraduate course in logic and axiomatics. In A decision method for elementary algebra and geometry Tarski showed that the first-order theory of the real numbers under addition and multiplication is decidable which is in contrast, in a way which is really surprising to non-experts, to the results of Gödel and Church who showed that the first-order theory of the natural numbers under addition and multiplication is undecidable. Cardinal Algebras presents a study of algebras satisfying certain properties which capture the arithmetic of cardinal numbers. In Undecidable theories Tarski showed that group theory, lattices, abstract projective geometry, closure algebras and others mathematical systems are undecidable. He had already made clear how pleased mathematicians should be that there is no solution to the general decision problem. In a lecture given at Harvard during 1939-40 Tarski said:-

... the solution of the decision problem in its most general form is negative. ... [Surely] many mathematicians experienced a profound feeling of relief when they heard of this result. Perhaps sometimes in their sleepless nights they thought with horror of the moment when some wicked metamathematician would find a positive solution, and design a machine which would enable us to solve any mathematical problem in a purely mechanical way.... The danger is now over ... [and] mathematicians ... can sleep quietly.

In Ordinal algebras Tarski defines an algebra which captures the properties of the additive theory of order types. It differs most strongly from the algebras which he presented in Cardinal algebras by having non-commutative addition.

The collected papers of Tarski were produced in four volumes edited by Steven R Givant and Ralph N McKenzie. John Corcoran, reviewing these volumes, writes:-

The mathematical community owes a debt of gratitude to Givant and McKenzie for their efforts in producing this invaluable collection of Alfred Tarski's works. It is only when we see Tarski's papers collected in one place that we can begin to appreciate the scope and profundity of his influence on modern mathematical thought and, in particular, on modern mathematical logic. Mathematical logic as we know it today is almost inconceivable without Tarski's contributions.

G H Moore writes in [1] described Tarski's character:-

Tarski was extroverted, quick-witted, strong-willed, energetic, and sharp-tongued. He preferred his research to be collaborative - sometimes working all night with a colleague - and was very fastidious about priority.

A B Feferman also describes aspects of his character writing in [25]:-

A charismatic leader and teacher, known for his brilliantly precise yet suspenseful expository style, Tarski had intimidatingly high standards for students, but at the same time he could be very encouraging, and particularly so to women - in contrast to the general trend. Some students were frightened away, but a circle of disciples remained, many of whom became world-renowned leaders in the field.

Tarski was honoured by being elected to the National Academy of Sciences, the Royal Netherlands Academy of Sciences, and the British Academy. He was made honorary editor of Algebra Universalis and served as President of the Association for Symbolic Logic from 1944 to 1946 and the International Union for the History and Philosophy of Science in 1956-57. He received honorary degrees from the Catholic University of Chile (1975) and the University of Marseilles (1977). In 1981 Berkeley awarded him the Berkeley Citation.