- Thoralf Skolem
Infobox_Scientist

name = Thoralf Skolem

birth_date = birth date|1887|5|23|mf=y

birth_place =Sandsvaer ,Buskerud ,Norway

residence =

nationality =

death_date = death date and age|1963|3|23|1887|5|23|mf=y

death_place =Oslo ,Norway

field =Mathematician

work_institution =Oslo University Christian Michelsen's Institute

alma_mater =Oslo University

doctoral_advisor =Axel Thue

doctoral_students =Øystein Ore

known_for =Skolem-Noether theorem

prizes =

religion =**Thoralf Albert Skolem**(May 23 ,1887 –March 23 ,1963 ) (IPA2|ˈtɔɾɑlf ˈskuləm) was a Norwegianmathematician known mainly for his work onmathematical logic andset 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 renamedOslo ), passing the university entrance examinations in 1905. He then enteredKristiania University to study mathematics, also taking courses inphysics ,chemistry ,zoology andbotany .In 1909, he began working as an assistant to the physicist

Kristian Birkeland , known for bombarding magnetized spheres withelectron s 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 thezodiacal light . He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center inmathematical logic ,metamathematics , andabstract 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 inChr. 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 andmathematical logic . He mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is theSkolem-Noether theorem , characterizing theautomorphisms of simple algebras. Skolem published a proof in 1927, butEmmy Noether independently rediscovered it a few years later.Skolem was among the first to write on

lattice s. In 1912, he was the first to describe a freedistributive lattice generated by "n" elements. In 1919, he showed that everyimplicative lattice (now also called aSkolem 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 theLöwenheim-Skolem theorem , which states that if a first-order theory has a model, then it has a countable model. His 1920 proof employed theaxiom of choice , but he later (1922 and 1928) gave proofs usingKönig'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 theoristsCharles Peirce andErnst Schroder , including ∏, ∑ as variable-binding quantifiers, in contrast to the notations ofPeano ,Principia Mathematica , and "Principles of Theoretical Logic ". In 1933 and later, Skolem pioneered the construction ofnon-standard model s 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 asSkolem'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 offinitism in mathematics. Skolem (1923) sets out hisprimitive recursive arithmetic , a very early contribution to the theory ofcomputable function s, 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 byprimitive recursion , then devising another system to prove properties of the objects defined by the first system. These two systems enabled him to defineprime number s 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 wasconsistent , complete, anddecidable . 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).

**ee also***

Model theory

*Skolem normal form

*Skolem's paradox

*Skolem sequence **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," 252–263.

**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***MacTutor: [

*http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Skolem.html Thoralf Skolem.*]

*MathGenealogy|18237

* Fenstad, Jens Erik, 1996, " [*http://www.hf.uio.no/ifikk/filosofi/njpl/vol1no2/skobio/node1.html 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=

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**Thoralf Skolem**— Thoralf Albert Skolem (23 mai 1887 23 mars 1963) était un mathématicien et logicien norvégien. Il est particulièrement connu pour les travaux en logique mathématique et théorie des ensembles qui portent à présent son nom, comme le théorème de… … Wikipédia en Français**Thoralf Skolem**— Albert Thoralf Skolem (* 23. Mai 1887 in Sandsvaer; † 23. März 1963 in Oslo) war ein norwegischer Mathematiker, Logiker und Philosoph. Seine Arbeiten lieferten grundlegende Resultate zur mathematischen Logik, insbesondere zu den Bereichen… … Deutsch Wikipedia**Albert Thoralf Skolem**— (* 23. Mai 1887 in Sandsvaer; † 23. März 1963 in Oslo) war ein norwegischer Mathematiker, Logiker und Philosoph. Seine Arbeiten lieferten grundlegende Resultate zur mathematischen Logik, insbesondere zu den Bereichen Modelltheorie und… … Deutsch Wikipedia**Skolem**— Thoralf Skolem Thoralf Albert Skolem (23 mai 1887 23 mars 1963) était un mathématicien et logicien norvégien. Il est particulièrement connu pour les travaux en logique mathématique et théorie des ensembles qui portent à présent son nom, comme le… … Wikipédia en Français**Skolem's paradox**— is the mathematical fact that every countable axiomatisation of set theory in first order logic, if consistent, has a model that is countable, even if it is possible to prove, from those same axioms, the existence of sets that are not countable.… … Wikipedia**Thoralf**— ist ein skandinavischer männlicher Vorname, abgeleitet vom altnordischen Þórr (Thor), der auch in Deutschland vorkommt.[1] Alternative Formen des Namens sind Toralf [2] oder Toralv. Inhaltsverzeichnis 1 Bekannte Namensträger 1.1 … Deutsch Wikipedia**Skolem**— Albert Thoralf Skolem (* 23. Mai 1887 in Sandsvaer; † 23. März 1963 in Oslo) war ein norwegischer Mathematiker, Logiker und Philosoph. Seine Arbeiten lieferten grundlegende Resultate zur mathematischen Logik, insbesondere zu den Bereichen… … Deutsch Wikipedia**Skolem-Paradox**— Das Löwenheim Skolem Theorem besagt, dass eine Menge von Aussagen der Prädikatenlogik erster Stufe, die in einem Modell mit einer überabzählbar unendlich großen Domäne erfüllt ist, immer auch in einem Modell mit einer abzählbar unendlich großen… … Deutsch Wikipedia**Skolem–Noether theorem**— In mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in ring theory which characterizes the automorphisms of simple rings. The theorem was first published by Skolem in 1927 in his paper… … Wikipedia**Skolem-Normalform**— Skolemform ist ein Begriff der Prädikatenlogik und bezeichnet eine prädikatenlogische Formel, die sich in einer Normalform nach Albert Thoralf Skolem befindet. Für Formeln in Skolemform existiert ein berechenbarer Test auf Erfüllbarkeit. Dies ist … Deutsch Wikipedia