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

- 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.
- 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.
- 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.
- 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.
- 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.
- Skolem was remarkably productive publishing around 180 papers on topics such as Diophantine equations, mathematical logic, group theory, lattice theory and set theory.
- One example is the Skolem-Noether theorem.
- These results were rediscovered by other mathematicians in the 1930s and in 1936 Skolem published Über gewisse 'Verbände' oder 'Lattices' Ⓣ(On some 'associations' or '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.
- 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.
- Skolem's position is more solid than that.
- 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 Ⓣ(Justification of elementary arithmetic by the recurrent mindset without the use of apparent variables with infinite extension area).
- 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 Ⓣ(On the theory of associative number systems).
- Skolem was president of the Norwegian Mathematical Society and an editor of the Norsk Matematisk Tidsskrift (The Norwegian Mathematical Journal) for many years.
- In 1957 Skolem retired but continued to produce top quality research.

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

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Origin Norway

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive