Citizendia
Your Ad Here

Thoralf Skolem
Image:Thoralf Skolem.jpg
Thoralf Albert Skolem (1887–1963)
Born May 23, 1887 (1887-05-23)
Sandsvaer, Buskerud, Norway
Died March 23, 1963 (aged 75)
Oslo, Norway
Residence Norway
Nationality Norwegian
Fields Mathematician
Institutions Oslo University
Christian Michelsen's Institute
Alma mater Oslo University
Doctoral advisor Axel Thue
Doctoral students Øystein Ore
Known for Skolem-Noether theorem

Thoralf Albert Skolem (May 23, 1887 – March 23, 1963) was a Norwegian mathematician known mainly for his work on mathematical logic and set theory. Events 1430 - Siege of Compiègne: Joan of Arc is captured by the Burgundians while leading an army to relieve Compiègne Year 1887 ( MDCCCLXXXVII) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Common Sandsvær is a former municipality in Buskerud county Norway. is a county in Norway, bordering Akershus, Oslo, Oppland, Sogn og Fjordane, Hordaland, Telemark, and Norway ( Norwegian: Norge ( Bokmål) or Noreg ( Nynorsk) officially the Kingdom of Norway, is a Constitutional Events 1174 - Jocelin, Abbot of Melrose, is elected Bishop of Glasgow. Year 1963 ( MCMLXIII) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. (called Christiania from 1624 to 1878 and Kristiania from 1878 to 1924 is the Capital and largest city of Norway. Norway ( Norwegian: Norge ( Bokmål) or Noreg ( Nynorsk) officially the Kingdom of Norway, is a Constitutional Norway ( Norwegian: Norge ( Bokmål) or Noreg ( Nynorsk) officially the Kingdom of Norway, is a Constitutional Norway ( Norwegian: Norge ( Bokmål) or Noreg ( Nynorsk) officially the Kingdom of Norway, is a Constitutional A mathematician is a person whose primary area of study and research is the field of Mathematics. The University of Oslo (Universitetet i Oslo Universitas Osloensis is the oldest and largest University in Norway, situated in the Norwegian capital Alma mater is Latin for "nourishing mother" It was used in Ancient Rome as a title for the mother Goddess, and in Medieval The University of Oslo (Universitetet i Oslo Universitas Osloensis is the oldest and largest University in Norway, situated in the Norwegian capital A doctorate is an Academic degree that indicates the highest level of academic achievement Axel Thue ( 19 February 1863 – 7 March 1922) was a Norwegian Mathematician, known for highly original work in Diophantine Øystein Ore ( 7 October 1899 in Oslo, Norway &ndash 13 August 1968 in Oslo was a Norwegian Mathematician In Mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in Ring theory which characterizes Events 1430 - Siege of Compiègne: Joan of Arc is captured by the Burgundians while leading an army to relieve Compiègne Year 1887 ( MDCCCLXXXVII) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Common Events 1174 - Jocelin, Abbot of Melrose, is elected Bishop of Glasgow. Year 1963 ( MCMLXIII) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Norway ( Norwegian: Norge ( Bokmål) or Noreg ( Nynorsk) officially the Kingdom of Norway, is a Constitutional A mathematician is a person whose primary area of study and research is the field of Mathematics. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic.

Contents

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. (called Christiania from 1624 to 1878 and Kristiania from 1878 to 1924 is the Capital and largest city of Norway. (called Christiania from 1624 to 1878 and Kristiania from 1878 to 1924 is the Capital and largest city of Norway. He then entered Kristiania University to study mathematics, also taking courses in physics, chemistry, zoology and botany. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Zoology (from Greek ζῷον, zoon, "animal" + λόγος, " Logos " "knowledge" is the branch of Botany, plant science(s, phytology, or plant biology is a branch of Biology and is the scientific study of plant Life

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. Kristian Olaf Birkeland ( December 13, 1867 - June 15, 1917) was born in Christiania ( Oslo today and wrote his first The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J 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. The zodiacal light is a faint roughly triangular whitish glow seen in the night sky which appears to extend up from the vicinity of the Sun along the Ecliptic 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. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In general metamathematics or meta-mathematics is a scientific reflection and Knowledge about mathematics seen as an entity/ object in Human Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules 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. Axel Thue ( 19 February 1863 – 7 March 1922) was a Norwegian Mathematician, known for highly original work in Diophantine

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. The University of Oslo (Universitetet i Oslo Universitas Osloensis is the oldest and largest University in Norway, situated in the Norwegian capital The Chr Michelsen Institute (CMI was founded in 1930 and it is currently the largest centre for development research in Scandinavia is the second largest city in Norway. It is located on the south-western coast of Norway in the county of Hordaland in between a group of mountains known as De syv fjell 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. Øystein Ore ( 7 October 1899 in Oslo, Norway &ndash 13 August 1968 in Oslo was a Norwegian Mathematician

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. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical 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. In Mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in Ring theory which characterizes In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and

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 Mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other 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 Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models 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. Leopold Löwenheim (1878 Krefeld Germany - 1957 Berlin) was a German Mathematician, known for his work in Mathematical logic. In Mathematical logic, the Löwenheim–Skolem theorem states that if a countable first-order theory has an infinite model then for every infinite Cardinal number 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. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. König's lemma or König's infinity lemma is a Theorem in Graph theory due to Dénes Kőnig (1936 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. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher For the actor see Ernst Schröder (actor. Ernst Schröder ( 25 November, 1841 Mannheim Germany – Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert 's and Wilhelm Ackermann 's classic text In 1933 and later, Skolem pioneered the construction of non-standard models of arithmetic and set theory. See also Formal interpretation In Model theory, a discipline within Mathematical logic, a non-standard model is a

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. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science 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. Skolem's paradox is the mathematical fact that every countable Axiomatisation of Set theory in First-order logic, if Consistent, has

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. Gödel's completeness theorem is a fundamental theorem in Mathematical logic that establishes a correspondence between semantic truth and syntactic provability in First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert 's and Wilhelm Ackermann 's classic text In any event, Kurt Gödel first proved this completeness in 1930. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher

Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In the Philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed 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. Primitive recursive arithmetic, or PRA, is a Quantifier -free formalization of the natural numbers Computable functions are the basic objects of study in computability theory. 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. The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict Subset of the recursive These two systems enabled him to define prime numbers and to set out a considerable amount of number theory. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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. Presburger arithmetic is the first-order theory of the Natural numbers with Addition, named in honor of Mojżesz Presburger, who published it In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural 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. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most

Hao Wang praised Skolem's work as follows:

"Skolem tends to treat general problems by concrete examples. Wang Hao, also Hao Wang ( 20 May 1921 &ndash 13 May 1995) was a Chinese American Logician, Philosopher 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). Wang Hao, also Hao Wang ( 20 May 1921 &ndash 13 May 1995) was a Chinese American Logician, Philosopher

References

Primary:

Writings in English translation:

Secondary:

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
Events 1430 - Siege of Compiègne: Joan of Arc is captured by the Burgundians while leading an army to relieve Compiègne Year 1887 ( MDCCCLXXXVII) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Common Events 1174 - Jocelin, Abbot of Melrose, is elected Bishop of Glasgow. Year 1963 ( MCMLXIII) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar.
© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic