# Thoralf Albert Skolem

### Quick Info

Born
23 May 1887
Sandsvaer, Buskerud, Norway
Died
23 March 1963
Oslo, Norway

Summary
Thoralf Skolem was a Norwegian mathematician who worked in mathematical logic and set theory.

### Biography

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.

### References (show)

1. H Oettel, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. J E Fenstadt (ed.), Thoralf Skolem, Selected works in logic (Oslo, 1970).
3. W Boos, Thoralf Skolem, Hermann Weyl and 'Das Gefühl der Welt als begrenztes Ganzes', in From Dedekind to Gödel, Boston, MA, 1992 (Dordrecht, 1995), 283-329.
4. R Crespo, Th Skolem (Spanish), Gaceta Mat. (1) 4 (1952), 109-112.
5. J E Fenstadt, Thoralf Albert Skolem in Memoriam, Nordisk Mathematisk Tidsskrift 45 (1963), 145-153.
6. J E Fenstadt, Thoralf Albert Skolem 1887-1963 : a biographical sketch, Nordic J. Philos. Logic 1 (2) (1996), 99-106.
7. G Gjone, Über Leben und Werk von Thoralf Skolem, Contributions to the history, philosophy and methodology of mathematics, Wiss. Z. Greifswald, Ernst- Moritz- Arndt- Univ. Math.-Natur. Reihe 33 (1-2) (1984), 19-21.
8. I Jané, Reflections on Skolem's relativity of set-theoretical concepts, Philos. Math. (3) 9 (2) (2001), 129-153.
9. H R Jervell, Thoralf Skolem: pioneer of computational logic, Nordic J. Philos. Logic 1 (2) (1996), 107-117.
10. W Ljunggren, Thoralf Albert Skolem in memoriam, Math. Scand. 13 (1963), 5-8.
11. G Mints, Thoralf Skolem and the epsilon substitution method for predicate logic, Nordic J. Philos. Logic 1 (2) (1996), 133-146.
12. T Nagell, Thoralf Skolem in Memoriam, Acta Mathematica 110 (1963), i-xi.
13. S Selberg, Thoralf Albert Skolem (Norwegian), Norske Vid. Selsk. Forh. (Trondheim) 36 (1963), 165-168.
14. H Wang, Skolem and Gödel, Nordic J. Philos. Logic 1 (2) (1996), 119-132.
15. H Wang, Skolem and Gödel, in Mathematics, education and philosophy (London, 1994), 184-193.