Thoralf Albert Skolem (23
May 1887 – 23 March 1963) ( ) was a Norwegian mathematician known
mainly for his work on mathematical
logic and set theory.
Life
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
Det Kongelige Frederiks
Universitet 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 Det Kongelige Frederiks Universitet. 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 Det kongelige Frederiks Universitet (renamed
the University of
Oslo in 1939) 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.
For more on Skolem's life, see Fenstad (1970).
Mathematics
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
lattice. 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 an infinite 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 Sanders Peirce and
Ernst Schroder, including ∏, ∑ as
variable-binding quantifiers, in contrast to the notations of
Peano,
Principia Mathematica, and
Principles of
Mathematical Logic. Skolem (1934) pioneered the
construction of
non-standard
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 Mathematical
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)
For more on Skolem's accomplishments, see
Hao Wang (1970).
See also
References
Primary:
- Skolem, Th. (1934) Über die Nicht-charakterisierbarkeit der
Zahlenreihe mittels endlich oder abzählbar unendlich vieler
Aussagen mit ausschliesslich Zahlenvariablen. Fundam. Math. 23,
150-161.
- 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," 252–263.
- 1922. "Some remarks on axiomatized set theory," 290-301.
- 1923. "The foundations of elementary arithmetic," 302-33.
- 1928. "On mathematical logic," 508–524.
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).
External links