陶拉尔夫·斯科伦Thoralf Skolem

陶拉尔夫·斯科伦 的父母是 Helene Olette Vaal 和 Even 斯科伦，后者是小学教师。尽管他的父亲是教师，斯科伦 来自一个农民家庭，他的大多数亲戚都是农民。他上了中学，于 1905 年在 Kristiania（后更名为 Oslo）参加了期末考试 Examen artium。然后他进入 Kristiania 大学学习数学，但也修了物理、化学、动物学和植物学的课程。

1909 年，斯科伦 担任物理学家 Kristian Birkeland 的助手，后者因用电子轰击磁化球体获得类似极光效果的实验而闻名，斯科伦 的第一批出版物是与 Birkeland 合写的物理学论文。斯科伦 于 1913 年参加国家考试，以优异成绩通过。他的学位论文 Undersokelser innenfor logikkens algebra Ⓣ（《逻辑代数研究》）被认为非常出色，以至于他的成就被报告给了挪威国王。然而，他继续担任 Birkeland 的助手，并于 1913 年与他一起前往苏丹观察黄道光。尽管作为 Birkeland 的助手从事物理学工作，他仍继续数学研究，在此期间他证明了关于格的显著结果，我们将在下面提到。1915 年，他前往哥廷根，在那里度过了冬季学期。当然，这是在第一次世界大战期间，哥廷根的条件极其困难。1916 年，他回到克里斯蒂安尼亚，被任命为大学的研究员。然而，正如 Fenstadt 在 [5] 中所解释的，他并没有正式攻读博士学位：-

……维戈·布朗 和 斯科伦 一致认为他们两人都不会费心去获得博士学位，大概觉得在挪威，这对年轻科学家的教育没有实际用处。

斯科伦于1918年在克里斯蒂安尼亚获得数学讲师（Docent），同年当选为挪威科学与文学院。尽管他早先与维戈·布朗意见一致，但他还是在1926年决定提交学位论文以获取博士学位[5]：-

……在二十世纪二十年代中期，年轻一代的挪威数学家崭露头角。看来斯科伦当时也觉得他应该满足拥有博士学位的正式要求，并且他从维戈·布朗那里“获得了许可”来提交学位论文。1924年，维戈·布朗曾是挪威理工学院数学教授。

斯科伦在克里斯蒂安尼亚（或1925年更名为奥斯陆）的导师是阿克塞尔·图厄，尽管后者已于1922年去世，即在斯科伦决定提交学位论文的四年前。论文题为Einige Sätze über ganzzahlige Lösungen gewisser Gleichungen und Ungleichungen Ⓣ（关于某些方程和不等式的整数解的若干定理），内容涉及某些代数方程和不等式的整数解。

1927年5月23日，斯科伦与伊迪丝·威廉敏娜·哈斯沃尔德结婚。他继续在奥斯陆大学工作，直到1930年，他搬到卑尔根的克里斯蒂安·米歇尔森研究所担任研究助理。虽然这个头衔听起来并不特别显赫，但实际上这是一个高级职位，斯科伦能够进行独立研究，没有任何行政或教学职责。然而，这份工作的一个条件是他必须住在卑尔根，这有一个缺点，即他无法接触到数学文献。他在卑尔根工作到1938年，当时51岁的他回到奥斯陆，担任大学数学教授。芬斯塔特在[5]中写道：-

[斯科伦]定期讲授代数和数论的研究生课程，很少讲授数理逻辑。[他]天性非常谦逊和内向。他没有创建任何学派，也没有研究生，但通过他的巨大成就和研究动力，他激励了不止一位年轻的挪威数学家。

斯科伦异常多产，发表了约180篇论文，主题包括丢番图方程、数理逻辑、group theory、格论和集合论。然而，正如芬斯塔特在[2]中解释的那样：-

斯科伦的大部分论文发表在挪威的期刊上，国外的数学家并不总能轻易获得这些论文。这导致后来其他人重新发现了他的结果。一个例子是斯科伦-Noether定理。

他在格论方面做了一些早期工作。例如，1912年，他第一个描述了由$n$个元素生成的自由分配格。他还在1919年证明了每个蕴涵格都是分配的，并且作为部分逆命题，每个有限分配格都是蕴涵格。这些结果在20世纪30年代被其他数学家重新发现，1936年斯科伦发表了Über gewisse 'Verbände' oder 'Lattices' Ⓣ（关于某些“关联”或“格”），这是对他1912年和1919年论文中自己结果的综述。

斯科伦扩展了利奥波德·勒文海姆的工作（1915年发表），给出了利奥波德·勒文海姆-斯科伦定理，并于1920年发表。该定理指出，如果一阶谓词演算中的理论有模型，那么它有一个可数模型。他1920年对这个结果的证明使用了选择公理，但后来在1922年和1928年，他使用约翰·萨穆埃尔·柯尼希引理（归功于朱利叶斯·科尼格）给出了不需要选择公理的证明。

他对恩斯特·策梅洛的公理集合论进行了改进，在1922年和1929年发表了工作。第一个是讲座Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre Ⓣ（《关于集合论公理辩护的若干评论》）的发表版本，他于1922年在第五届斯堪的纳维亚数学大会上作了该讲座。在这里，他应用勒文海姆-斯科伦定理来展示后来被称为斯科伦悖论的内容：如果恩斯特·策梅洛的集合论公理系统是一致的，那么它必须在一个可数域内可满足。

Jané在[8]中写道：-

斯科伦通常被描绘成主张某些本来很好理解的概念仅仅因为无法在一阶语言中刻画就可疑；特别是，由于集合论的所有一阶形式化（如果一致）都有可数模型，不可数性的概念是有缺陷的。……斯科伦的立场比这更坚实。我认为斯科伦是在主张，为不可数集合的存在性提供的所有证据都是不确定的，而他坚持考虑可数模型的原因是，当时公理化被提出来作为确保集合论的唯一方式，并且集合是什么以及哪些集合存在被声称是由公理及其模型决定的（就像欧几里得几何是关于什么被声称是由大卫·希尔伯特的公理及其模型决定的一样）。在这种情况下，引入可数模型是完全合理的，尤其是因为如果没有集合论的手段，就无法提供其他模型。今天我们可能不再支持这一主张，但如果我们确实相信存在不可数集合，我们就应该愿意遵从斯科伦的要求，即它们的存在性应由某种手段而非仅仅形式公设来证实。

1923年，斯科伦还在他的论文Begründung der elementären Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnugsbereich Ⓣ（通过递归思维模式而不使用具有无限扩展区域的表观变量来证明初等算术）中发展了递归函数理论，作为避免所谓无限悖论的一种手段。在其中，他使用两个系统发展了数论，一个通过原始递归定义对象，另一个系统证明由第一个系统定义的对象性质。用这些他定义了素数并发展了相当多的数论。Jervell在[9]中认为斯科伦是计算机科学的先驱：-

斯科伦的两个系统可以被视为一种用于定义对象的编程语言和一种用于证明对象性质的编程逻辑。

从1933年起，他在元逻辑方面做了开创性工作，并构造了算术的非标准模型。Hao 王宪钟在A survey of Skolem's work in logic中写道，该文收录于[2]：-

如果要挑出一项最引人入胜的成果，那大概就是他在集合论和数论的非标准模型方面的工作。

王宪钟还指出，阅读斯科伦的原始论文[2]是多么有用：-

斯科伦有一种通过具体例子来处理一般问题的倾向。证明的呈现顺序往往似乎与他发现它们的顺序相同。

我们上面提到斯科伦研究代数，也提到了斯科伦-Noether定理。斯科伦于1927年在一篇论文Zur Theorie der assoziativen ZahlensystemeⓉ（论结合数系的理论）中发表了这一定理。它刻画了单代数的自同构，后来被埃米·诺特重新发现。

斯科伦曾任挪威数学会主席，并多年担任Norsk Matematisk Tidsskrift（《挪威数学杂志》）的编辑。在Mathematica Scandinavica创办之后（关于该杂志创办的经过，见挪威数学会条目），他担任这份新杂志的编辑。他获得过许多荣誉，例如1954年被挪威国王授予圣奥拉夫皇家勋章一级骑士称号。1962年，他还在特隆赫姆获得了Det Kongelige Norske Vitenskabers Selskab颁发的Gunnerus奖章。

1957年，斯科伦退休，但继续产出顶尖质量的研究。此后数年，他多次前往美国。尽管斯科伦去世时已年近76岁，但他的去世完全出人意料，因为他仍然是一位极其活跃且高产的数学家[5]：-

年龄似乎并未削弱他的研究动力或创造能力。

他曾计划再次前往美国，并已接受邀请，将在那里的几所大学发表演讲。他的去世非常突然。

Thoralf Skolem's parents were Helene Olette Vaal and Even Skolem, who was primary school teacher. Although his father was a teacher, Thoralf came from a farming family with most of his relations being farmers. He attended secondary school taking the final examination, the Examen artium, in Kristiania (later renamed Oslo) in 1905. He then entered Kristiania University to study mathematics, but he also took courses on physics, chemistry, zoology and botany.

In 1909 Skolem took a job as assistant to the physicist Kristian Birkeland, who was famed for his experiments with the aurora-like effect obtained by bombarding a magnetized sphere with electrons, and Skolem's first publications were physics papers written jointly with Birkeland. Skolem took his state examination in 1913, passing with distinction. His dissertation Undersokelser innenfor logikkens algebra Ⓣ was considered so outstanding that his achievement was reported to the King of Norway. He continued to act as Birkeland's assistant, however, and travelled with him to the Sudan in 1913 to observe the zodiacal light. Despite working on physics as Birkeland's assistant, he continued mathematical research and during this time he proved notable results on lattices which we mention below. In 1915 he travelled to Göttingen where he studied during the winter semester. Of course this was during the years of World War I, and conditions in Göttingen were extremely difficult. In 1916 he returned to Kristiania where he was appointed as a research fellow at the university. He did not, however, formally study for a doctorate as Fenstadt explains in [5]:-

... Viggo Brun and Skolem agreed that neither of them would bother to obtain the degree of Doctor, probably feeling that, in Norway, it served no useful function in the education of a young scientist.

Skolem became a Docent in Mathematics in Kristiania in 1918, and in the same year he was elected to the Norwegian Academy of Science and Letters. Despite his earlier agreement with Viggo Brun, he decided to submit a thesis for a doctorate in 1926 [5]:-

... in the middle twenties a younger generation of Norwegian mathematician emerged. It seems that Skolem then felt he too ought to fulfil the formal requirement of having a doctorate, and he "obtained permission" from Brun to submit a thesis. In 1924 Brun had been a professor in mathematics at the Norwegian Institute of Technology.

Skolem's advisor in Kristiania (or Oslo as it was renamed in 1925) had been Axel Thue although he had died in 1922, four years before Skolem decided to submit his thesis. It was entitled Einige Sätze über ganzzahlige Lösungen gewisser Gleichungen und Ungleichungen Ⓣ, and was on integral solutions of certain algebraic equations and inequalities.

On 23 May 1927 Skolem married Edith Wilhelmine Hasvold. He continued to work at the University of Oslo until 1930 when he moved to the Christian Michelsen's Institute in Bergen as a Research Associate. Although this does not sound a particularly grand title, in fact the post was a senior one in which Skolem was able to conduct independent research without any administrative or teaching duties. A condition of the job, however, was that he had to live in Bergen and that had the disadvantage that he did not have access to mathematical literature. He worked in Bergen until 1938 when, at the age of 51, he returned to Oslo as Professor of Mathematics at the university. Fenstadt writes in [5]:-

[Skolem] conducted the regular graduate courses in algebra and number theory, and rather infrequently lectured on mathematical logic. [He] was very modest and retiring by nature. He did not create any school and had no research students, but through his great accomplishments and research drive he inspired more than one of the younger Norwegian mathematicians.

Skolem was remarkably productive publishing around 180 papers on topics such as Diophantine equations, mathematical logic, group theory, lattice theory and set theory. However, as Fenstadt explains in [2]:-

Skolem published ... most of his papers in Norwegian journals, and they have not always been easy to obtain for mathematicians abroad. This has had the consequence that others later have rediscovered his results. One example is the Skolem-Noether theorem.

He did some early work in lattice theory. For example in 1912 he was the first to give a description of a free distributive lattice generated by $n$ elements. He also showed in 1919 that every implicative lattice is distributive and, as a partial converse, that every finite distributive lattice is implicative. These results were rediscovered by other mathematicians in the 1930s and in 1936 Skolem published Über gewisse 'Verbände' oder 'Lattices' Ⓣ which is a survey of his own results from the 1912 and 1919 papers.

Skolem extended work by Löwenheim (published in 1915) to give the Löwenheim-Skolem theorem, which he published in 1920. It states that if a theory within first-order predicate calculus has a model then it has a countable model. His 1920 proof of this result used the axiom of choice, but later in 1922 and 1928 he gave proofs using König's lemma (due to Julius König) which do not require the axiom of choice.

He made refinements to Zermelo's axiomatic set theory, publishing work in 1922 and 1929. The first was the published version of the lecture Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre Ⓣ which he gave in 1922 at the 5th Scandinavian Mathematics Congress. Here he applied the Löwenheim-Skolem theorem to show what became known as Skolem's paradox: If the Zermelo's axiomatic system for set theory is consistent then it must be satisfiable within a countable domain.

Jané writes in [8]:-

Skolem is commonly portrayed as arguing that certain otherwise well understood concepts are suspect simply because they cannot be characterized in a first-order language; in particular that, since all first-order formalizations of set theory (if consistent) have countable models, the concept of uncountability is flawed. ... Skolem's position is more solid than that. I see Skolem as arguing that all the evidence that has been given for the existence of uncountable sets is inconclusive, and the reason why he insists on considering countable models is that axiomatisation was put forward at the time as the only way to secure set theory, and what sets are and which sets exist was claimed to be determined by the axioms and their models (much as what Euclidean geometry is about was claimed to be determined by Hilbert's axioms and their models). In this situation, bringing countable models into play was perfectly in order, all the more so as no other models could be supplied without set-theoretical means. Today we may no longer uphold this claim, but if we do believe that there are uncountable sets, we should be willing to comply with Skolem's requirement that their existence be substantiated by some means other than mere formal postulation.

In 1923 Skolem also developed a theory of recursive functions as a means of avoiding the so-called paradoxes of the infinite in his paper Begründung der elementären Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnugsbereich Ⓣ. In it he developed number theory using two systems, one to define objects by primitive recursion, the other system to prove properties of the objects defined by the first system. With these he defined prime numbers and developed a considerable amount of number theory. Jervell, in [9], sees Skolem as being a pioneer in computer science:-

Skolem's two systems could be considered as a programming language for defining objects and a programming logic for proving properties about the objects.

From 1933 he did pioneering work in metalogic and constructed a nonstandard model of arithmetic. Hao Wang, in A survey of Skolem's work in logic which appears in [2] writes:-

If one has to single out one most intriguing item, it would probably be his work on nonstandard models of set theory and number theory.

Wang also indicates how useful it is to read Skolem's original papers [2]:-

Skolem has a tendency of treating general problems by concrete examples. Often proofs seem to be presented in the same order as he came to discover them.

We mentioned above that Skolem worked on algebra, and we also mentioned the Skolem-Noether theorem. Skolem published this theorem in 1927 in a paper Zur Theorie der assoziativen Zahlensysteme Ⓣ. It characterizes the automorphisms of simple algebras and was later rediscovered by Emmy Noether.

Skolem was president of the Norwegian Mathematical Society and an editor of the Norsk Matematisk Tidsskrift (The Norwegian Mathematical Journal) for many years. After the creation of Mathematica Scandinavica (see the article on the Norwegian Mathematical Society for the story of the founding of this journal) he acted as an editor for the new journal. He received many honours such as being named a Knight of the First Class in the Royal Order of St Olav in 1954 by the King of Norway. He also received the Gunnerus Medal by Det Kongelige Norske Vitenskabers Selskab in 1962 in Trondheim.

In 1957 Skolem retired but continued to produce top quality research. He made several trips to the United States in the following years. Although Skolem was approaching 76 years of age when he died, his death was totally unexpected since he was still an extremely active and highly productive mathematician [5]:-

Age had not seemed to diminish either his research drive or creative ability.

He had planned a further trip to the United States and had accepted invitations to speak at several universities there. His death came very suddenly.