Thoralf Skolem
Thoralf Albert Skolem (
May 23,
1887 -
March 23,
1963) was a
Norwegian mathematician known mainly for his work on
mathematical logic and
set theory.
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 in 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. Although this job title does not sound impressive, the post was in fact a senior one, enabling 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.
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 not infrequently 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 aqre often pregnant and potentially capable of wide application. He was very much a 'free spirit': he did not belog 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)
Skolem is a curiously understudied founder of
mathematical logic and
set theory. His importance is not in doubt; the classic source book
Jean Van Heijenoort (1967) devotes more pages to Skolem than to anyone other than
Frege (who has the most pages because Van Heijenoort translated the entire
Begriffsschrifft). Also see Wang's essay included in Skolem (1970). Moreover, language is not a barrier to studying Skolem's work. While he did publish a fair bit in Norwegian, a language closed to most scholars, his heavily cited papers, including the four translated in Van Heijenoort (1967), were written in German, which should present no difficulty for a historian of mathematics or logic. After WWII, he often published in English.
Frankel's modification of Zermelo's axiomatic set theory has led to that theory being called
Zermelo-Fraenkel set theory (often abbreviated ZF or ZFC), but Skolem's equally important and independent contribution is only mentioned. Skolem is often cited where appropriate, but no one lingers over his work. According to the index of
Ivor Grattan-Guinness (2000), that otherwise magisterial study mentions or cites Skolem only four times; the only work by Skolem included in Grattan-Guinness's 76pp bibliography is Skolem (1923) (failing to mention that it can be found in Van Heijenoort) and one review. Likewise, the bibliography to Ewald (1996) includes only Skolem (1922), and the index includes only four entries under Skolem's name.
The only exception to this lack of detailed study of Skolem's work appears to be the monograph Brady (2000), who was mainly interested in Skolem as a founder of the
model theoretic outlook, by virtue of his contribution to the
Lowenheim-Skolem theorem. Skolem held unusual philosophical views that deserve closer study. He was a pioneer of
finitism, a fact consistent with his preference for
abstract algebra over analysis and
topology. Although he contributed importantly to the now-dominant ZFC consensus, he did not believe in
axiomatic set theory, pointing to his
eponymous paradox as the reason. For much of the 20th century, it could perhaps be said that Skolem was being sidelined in a manner resembling the fate of other pioneer model theorists such as
Charles Peirce and
Ernst Schroder, but with the postwar ascension of
model theory, thanks to the writings of
Jaakko Hintikka and of
Tarski and his students, this reason no longer applies. Hence the lack of discussion of Skolem's work remains unexplained.
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 orginally 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).
Other references:
*Ewald, W. B., ed., 1996.
From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Univ. Press.
*
Ivor Grattan-Guinness, 2000.
The Search for Mathematical Roots, 1870-1940. Princeton Univ. Press.
*
Löwenheim-Skolem theorem*
model theory*
Skolemization*
Skolem function*
Skolem normal form*
Skolem paradox*
Skolem-Noether theorem*MacTutor:
Thoralf Skolem.* Fenstad, Jens Erik, 1996, "
Thoralf Albert Skolem 1887-1963: A Biographical Sketch,"
Nordic Journal of Philosophical Logic 1: 99-106.
{{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=
|
