Thoralf Albert Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician known mainly for his work on mathematical logic
and set theory.
Life
On Skolem's life, see Fenstad (1970). Although Skolem's father was a primary school teacher, most of his extended family were
farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the
university entrance examinations in 1905. He then entered Kristiania University to study mathematics, also taking courses in
physics, chemistry, zoology
and botany.
In 1909, he began working as an assistant to the physicist Kristian Birkeland,
known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with
Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations
on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the
leading research center in mathematical logic, metamathematics, and abstract algebra, fields in which Skolem
eventually excelled. In 1916 he was appointed a research fellow at Kristiania University. In 1918, he became a Docent in
Mathematics and was elected to the Norwegian Academy of Science and Letters.
Skolem did not at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary in Norway. He later
changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations
and inequalities. His notional thesis advisor was Axel Thue, even though Thue had died in
1922.
In 1927, he married Edith Wilhelmine Hasvold.
Skolem continued to teach at Kristiania University (renamed the University of Oslo
in 1925) until 1930 when he became a Research Associate in Chr. Michelsen
Institute in Bergen. This senior post allowed Skolem to conduct research free of
administrative and teaching duties. However, the position also required that he reside in Bergen, a city which then lacked a
university and hence had no research library, so that he was unable to keep abreast of the mathematical literature. In 1938, he
returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the graduate courses in algebra
and number theory, and only occasionally on mathematical logic. Over the course of his entire career, he had but one Ph.D.
student, but that student was a splendid one, Øystein Ore, who went on to a career in the
USA.
Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift ("The
Norwegian Mathematical Journal") for many years. He was also the founding editor of Mathematica Scandinavica.
After his 1957 retirement, he made several trips to the United States, speaking and teaching at universities there. He
remained intellectually active until his sudden and unexpected death.
Mathematics
On Skolem's accomplishments, see Hao Wang (1970). Skolem published around 180 papers on
Diophantine equations, group theory,
lattice theory, and most of all, set theory and
mathematical logic. He mostly published in Norwegian journals with limited
international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem-Noether theorem, characterizing the automorphisms of
simple algebras. Skolem published a proof in 1927, but Emmy Noether independently
rediscovered it a few years later.
Skolem was among the first to write on lattices. In 1912, he was the first to describe a
free distributive lattice generated by n elements. In 1919, he showed that
every implicative lattice (now also called a Skolem lattice) is
distributive and, as a partial converse, that every finite distributive lattice is implicative. After these results were
rediscovered by others, Skolem published a 1936 paper in German, "Über gewisse 'Verbände' oder 'Lattices'", surveying his earlier
work in lattice theory.
Skolem was a pioneer model theorist. In 1920, he greatly simplified the proof of a
theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim-Skolem theorem, which states that if a first-order theory has a model, then it has a
countable model. His 1920 proof employed the axiom of choice, but he later (1922 and
1928) gave proofs using König's lemma in place of that axiom. It is notable that Skolem,
like Löwenheim, wrote on mathematical logic and set theory employing the notation of his fellow pioneering model theorists
Charles Peirce and Ernst Schroder, including ∏, ∑
as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica, and Principles of
Theoretical Logic. In 1933 and later, Skolem pioneered the construction of nonstandard models of arithmetic and set theory.
Skolem (1922) refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any
property that can be coded in first-order logic. The resulting axiom is now part of
the standard axioms of set theory. Skolem also pointed out that a consequence of the Löwenheim-Skolem theorem is what is now
known as Skolem's paradox: If Zermelo's axioms are consistent, then they must be
satisfiable within a countable domain, even though they prove the existence of uncountable sets.
The completeness of first-order
logic is an easy corollary of results Skolem proved in the early 1920s and discussed in Skolem (1928), but he failed to
note this fact, perhaps because mathematicians and logicians did not become fully aware of completeness as a fundamental
metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Theoretical Logic clearly articulated it. In any event,
Kurt Gödel first proved this completeness in 1930.
Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics. Skolem (1923) sets out his primitive
recursive arithmetic, a very early contribution to the theory of computable
functions, as a means of avoiding the so-called paradoxes of the infinite. Here he developed the arithmetic of the natural
numbers by first defining objects by primitive recursion, then devising
another system to prove properties of the objects defined by the first system. These two systems enabled him to define
prime numbers and to set out a considerable amount of number theory. If the first of these
systems can be considered as a programming language for defining objects, and the second as a programming logic for proving
properties about the objects, Skolem can be seen as an unwitting pioneer of theoretical computer science.
In 1929, Presburger proved that Peano
arithmetic without multiplication was consistent, complete, and
decidable. The following year, Skolem proved that the same was true of Peano arithmetic
without addition, a system named Skolem arithmetic in his honor. Gödel's famous 1931
result is that Peano arithmetic itself (with both addition and multiplication) is incompletable and hence a fortiori undecidable.
Hao Wang praised Skolem's work as follows:
"Skolem tends to treat general problems by concrete examples. He often seemed to present proofs in the same order as he came
to discover them. This results in a fresh informality as well as a certain inconclusiveness. Many of his papers strike one as
progress reports. Yet his ideas are often pregnant and potentially capable of wide application. He was very much a 'free spirit':
he did not belong to any school, he did not found a school of his own, he did not usually make heavy use of known results... he
was very much an innovator and most of his papers can be read and understood by those without much specialized knowledge. It
seems quite likely that if he were young today, logic... would not have appealed to him." (Skolem 1970: 17-18)
References
Primary:
- Skolem, T. A., 1970. Selected works in logic, Fenstad, J. E., ed. Oslo: Scandinavian University Books. Contains 22
articles in German, 26 in English, 2 in French, 1 English translation of an article originally published in Norwegian, and a
complete bibliography.
Writings in English translation:
- Jean van Heijenoort, 1967. From Frege to Gödel: A Source Book in Mathematical
Logic, 1879-1931. Harvard Univ. Press.
- 1920. "Logico-combinatorial investigations on the satisfiability or provability of mathematical propositions: A simplified
proof of a theorem by Loewenheim," 252-63.
- 1922. "Some remarks on axiomatized set theory," 290-301.
- 1923. "The foundations of elementary arithmetic," 302-33.
- 1928. "On mathematical logic," 508-24.
Secondary:
- Brady, Geraldine, 2000. From Peirce to Skolem. North Holland.
- Fenstad, Jens Erik, 1970, "Thoralf Albert Skolem in Memoriam" in Skolem (1970: 9-16).
- Hao Wang, 1970, "A survey of Skolem's work in logic" in Skolem (1970: 17-52).
See also
External links
| Persondata |
| NAME |
Skolem, Thoralf Albert |
| ALTERNATIVE NAMES |
|
| SHORT DESCRIPTION |
Norwegian mathematician |
| DATE OF BIRTH |
May 23, 1887 |
| PLACE OF BIRTH |
|
| DATE OF DEATH |
March 23, 1963 |
| PLACE OF DEATH |
|
pms:Thoralf Albert Skolem
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