31:
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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)
575:
298:, 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. Skolem's Ph.D. student
561:
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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
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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
458:
is a 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
382:) 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, "Ăber gewisse 'VerbĂ€nde' oder 'Lattices'", surveying his earlier work in lattice theory.
683:, 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.
241:-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
265:, 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
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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.
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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
405:
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
438:. The resulting axiom is now part of the standard axioms of set theory. Skolem also pointed out that a consequence of the LöwenheimâSkolem theorem is what is now known as
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1920. "Logico-combinatorial investigations on the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by Löwenheim," 252â263.
347:. He mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the
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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
442:: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets.
294:. This senior post allowed Skolem to conduct research free of administrative and teaching duties. However, the position also required that he reside in
488:, 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
379:
646:"Ăber die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzĂ€hlbar unendlich vieler Aussagen mit ausschlieĂlich Zahlenvariablen"
856:
397:, which states that if a countable first-order theory has an infinite model, then it has a countable model. His 1920 proof employed the
492:, then devising another system to prove properties of the objects defined by the first system. These two systems enabled him to define
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Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in
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516:. The following year, Skolem proved that the same was true of Peano arithmetic without addition, a system named
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524:'s famous 1931 result is that Peano arithmetic itself (with both addition and multiplication) is
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in 1939) until 1930 when he became a
Research Associate in Chr. Michelsen Institute in
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Some theorems about integral solutions to certain algebraic equations and inequalities
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Fenstad, Jens Erik, 1970, "Thoralf Albert Skolem in
Memoriam" in Skolem (1970: 9â16).
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Skolem served as president of the
Norwegian Mathematical Society, and edited the
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Skolem continued to teach at Det kongelige
Frederiks Universitet (renamed the
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Hao Wang, 1970, "A survey of Skolem's work in logic" in Skolem (1970: 17â52).
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182:; 23 May 1887 â 23 March 1963) was a Norwegian mathematician who worked in
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206:), passing the university entrance examinations in 1905. He then entered
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From Frege to Gödel: A Source Book in
Mathematical Logic, 1879â1931
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245:. He also traveled with Birkeland to the Sudan to observe the
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For more on Skolem's accomplishments, see Hao Wang (1970).
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In 1909, he began working as an assistant to the physicist
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of simple algebras. Skolem published a proof in 1927, but
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1923. "The foundations of elementary arithmetic," 302-33.
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1922. "Some remarks on axiomatized set theory," 290-301.
389:. In 1920, he greatly simplified the proof of a theorem
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For more on Skolem's academic life, see
Fenstad (1970).
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Thoralf Albert Skolem 1887-1963: A Biographical Sketch
687:
Skolem, Thoralf (23 April 2018). Zach, Richard (ed.).
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401:, but he later (1922 and 1928) gave proofs using
359:independently rediscovered it a few years later.
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366:. In 1912, he was the first to describe a free
233:, known for bombarding magnetized spheres with
427:. Skolem (1934) pioneered the construction of
283:In 1927, he married Edith Wilhelmine Hasvold.
249:. He spent the winter semester of 1915 at the
484:, a very early contribution to the theory of
253:, at the time the leading research center in
210:to study mathematics, also taking courses in
480:in mathematics. Skolem (1923) sets out his
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469:first proved this completeness in 1930.
374:elements. In 1919, he showed that every
267:Norwegian Academy of Science and Letters
801:Nordic Journal of Philosophical Logic 1
776:MacTutor History of Mathematics Archive
732:1928. "On mathematical logic," 508â524.
393:first proved in 1915, resulting in the
362:Skolem was among the first to write on
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465:clearly articulated it. In any event,
327:Skolem published around 180 papers on
243:Investigations on the Algebra of Logic
280:, even though Thue had died in 1922.
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857:20th-century Norwegian philosophers
208:Det Kongelige Frederiks Universitet
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540:praised Skolem's work as follows:
276:. His notional thesis advisor was
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429:non-standard models of arithmetic
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472:Skolem distrusted the completed
462:Principles of Mathematical Logic
424:Principles of Mathematical Logic
302:went on to a career in the USA.
708:Writings in English translation
476:and was one of the founders of
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482:primitive recursive arithmetic
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791:Mathematics Genealogy Project
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689:"Skolem's 1920, 1923 Papers"
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795:Fenstad, Jens Erik, 1996, "
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508:without multiplication was
307:Norsk Matematisk Tidsskrift
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827:Norwegian mathematicians
781:University of St Andrews
744:Brady, Geraldine, 2000.
395:LöwenheimâSkolem theorem
311:Mathematica Scandinavica
138:Chr. Michelsen Institute
108:LöwenheimâSkolem theorem
720:. Harvard Univ. Press.
677:Selected works in logic
668:10.4064/fm-23-1-150-161
654:Fundamenta Mathematicae
251:University of Göttingen
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16:Norwegian mathematician
832:Mathematical logicians
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407:Charles Sanders Peirce
349:SkolemâNoether theorem
104:SkolemâNoether theorem
746:From Peirce to Skolem
675:Skolem, T. A., 1970.
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419:Principia Mathematica
385:Skolem was a pioneer
351:, characterizing the
329:Diophantine equations
171:Thoralf Albert Skolem
767:Robertson, Edmund F.
486:computable functions
368:distributive lattice
837:Norwegian logicians
765:O'Connor, John J.;
714:Jean van Heijenoort
490:primitive recursion
378:(now also called a
376:implicative lattice
339:, and most of all,
604:Skolem normal form
345:mathematical logic
288:University of Oslo
255:mathematical logic
231:Kristian Birkeland
184:mathematical logic
842:Lattice theorists
599:Skolem arithmetic
589:Leopold Löwenheim
581:Philosophy portal
518:Skolem arithmetic
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436:first-order logic
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567:Biography portal
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531:a posteriori
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323:Mathematics
300:Ăystein Ore
159:Ăystein Ore
82:Nationality
48:23 May 1887
811:Categories
630:References
528:and hence
510:consistent
502:Presburger
467:Kurt Gödel
341:set theory
200:Kristiania
188:set theory
175:Norwegian:
44:1887-05-23
803:: 99-106.
739:Secondary
698:4 January
514:decidable
500:In 1929,
497:science.
278:Axel Thue
235:electrons
216:chemistry
149:Axel Thue
85:Norwegian
716:, 1967.
644:(1934).
553:See also
538:Hao Wang
478:finitism
474:infinite
364:lattices
76:, Norway
52:SandsvĂŠr
789:at the
635:Primary
220:zoology
212:physics
421:, and
296:Bergen
292:Bergen
261:, and
239:aurora
224:botany
121:Fields
56:Norway
649:(PDF)
522:Gödel
415:Peano
700:2024
450:The
409:and
343:and
222:and
204:Oslo
194:Life
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63:Died
38:Born
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