Albert Thoralf Skolem

Albert Thoralf Skolem (1930s)

Albert Thoralf Skolem (born May 23, 1887 in Sandsvaer , † March 23, 1963 in Oslo ) was a Norwegian mathematician , logician and philosopher .

His work provided fundamental results in mathematical logic , in particular in the areas of model theory and computability . But he also made important contributions to basic mathematical research such as predicate logic , class logic , recursion theory , set theory and the fundamentals of arithmetic , as well as in algebra and number theory .

biography

Skolem was a teacher's son and studied in Kristiania from 1905 (called Oslo from 1925). From 1909 he worked for the physicist Kristian Birkeland (known for his studies of the northern lights), with whom he also undertook an expedition to Sudan in 1913 . Skolem's dissertation Undersøkelserinnenfor logikkens algebra (Studies on the Algebra of Logic) received a lot of attention and was even reported to the King of Norway. In 1915 he traveled to Göttingen, where he studied during the winter semester. In 1916 he returned to Kristiania and took up a research position at the university under Axel Thue , where he initially agreed with Viggo Brun , who also worked there , not to work towards a doctorate. In 1918 Skolem became a lecturer in mathematics in Kristiania and in the same year became a member of the Norwegian Academy of Sciences .

In 1926 Skolem submitted a dissertation ( Some theorems about integer solutions of certain equations and inequalities ) on number theory (actually, he and his friend Viggo Brun had decided to forego it because they did not consider it necessary in Norway). His actual doctoral supervisor, the well-known number theorist Axel Thue , had died four years ago.

In 1927 Skolem married Edith Wilhelmine Hasvold and continued to work at the University of Oslo until he went to Bergen with his wife in 1930 to work as a researcher at the Christian Michelsen Institute. He worked there until 1938, when he accepted a call to Oslo and took over a chair for mathematics there, which he held until his retirement in 1957. He only gave occasional lectures on his actual field of mathematical logic and did not establish a school in Norway. Since he mostly published in Norwegian magazines, some of his findings went unnoticed until others rediscovered them. For example, he wrote an essay on the theory of associations as early as 1912 and characterized the automorphisms of simple algebras in 1927, which was later rediscovered by Emmy Noether (Skolem-Noether theorem). Skolem remained scientifically active until his death.

In 1954 Skolem was knighted by the Norwegian king. In 1962 he received the Gunnerus Medal from the Royal Norwegian Society of Sciences. He was president of the Norwegian Mathematical Society and long-time editor of Norsk Matematisk Tidsskrift and Mathematica Scandinavica . In 1936 he was invited speaker at the International Congress of Mathematicians in Oslo ( A Comment on the Decision Problem ) and in 1950 in Cambridge (Massachusetts) (Remarks on the foundation of set theory). In 1962 he gave a lecture at the ICM in Stockholm ( A theorem on recursively enumerable sets ).

His PhD students include Øystein Ore and Wilhelm Ljunggren .

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By means of the predicate logic normal form ( Skolem form ) named after him, he gave a clear proof in 1920 for the Löwenheim theorem (1915) that every satisfiable expression of the first order predicate calculus can be fulfilled in an at most countable range, so that this theorem is today theorem von Löwenheim and Skolem is mentioned. Skolem also pointed out in 1922 the apparently paradoxical consequences of this theorem in axiomatic set theory (" Skolem Paradox ").

In 1929 he gave the first precise predicate logic formalization of the Zermelo-Fraenkel set theory . Skolem put the final point in the axiomatization of set theory by using formalization to give the axiom of comprehension its customary version today. The term commonly used today, the primitive recursive function , goes back to Skolem (1923).

He showed that Peano arithmetic cannot be finally axiomatized. Skolem also made a number of contributions to the decision problem . He made the first attempt to build an axiomatic set theory with an unrestricted comprehension axiom on the basis of a multi-valued logic .

In 1933 he constructed a non-standard model of arithmetic .

In the 1930s he developed a p-adic method for solving a large class of Diophantine equations (Skolem's method). It has long been one of the few general solution methods applicable to larger classes of Diophantine equations.

In the field of algebra he published in 1927 a theorem known today as the Skolem-Noether theorem, according to which two embeddings of a simple algebra  in a central simple algebra  differ only in the conjugation with an invertible element : ${\displaystyle f,g\colon A\to B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle b\in B}$

${\displaystyle f(x)=b^{-1}\cdot g(x)\cdot b\quad {\mbox{für alle }}x\in A}$.

This result was independently proven by Emmy Noether .

Fonts

• Jens Erik Fenstad (Editor) Thoralf Skolem. Selected Works in Logic , Oslo, University Press 1970
• Investigations into the axioms of the class calculus and into the problems of production and summation which concern certain classes of propositions , 1919
• Logical-combinatorial studies on the satisfiability and provability of mathematical theorems together with a theorem about dense sets , 1920
• Some remarks on the axiomatic justification of set theory , 1922–1923
• Justification of elementary arithmetic through the recurrent way of thinking without the application of apparent variables with an infinite range of expansion , 1923
• On the theory of associative number systems , 1927
• On some fundamental questions in mathematics , 1929
• On the Fundamental Discussion in Mathematics , 1929–1930
• About some sentence functions in arithmetic , 1930–1931
• About the impossibility of a complete characterization of the series of numbers by means of a finite system of axioms , 1933
• About the non-characterisability of the series of numbers by means of a finite or countable infinite number of statements with exclusively number variables , 1934
• About the satisfiability of certain counting expressions , 1935
• About the traceability of some relations defined by recursions to 'arithmetic' , 1936–1937
• Sur la porteé de Löwenheim-Skolem , 1938
• Some remarks on the induction schemes in recursive number theory , 1939
• Some remarks on recursive arithmetic , 1944
• Remarks on the axiom of comprehension , 1957

Web links

• Literature by and about Albert Thoralf Skolem in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Albert Thoralf Skolem. In: MacTutor History of Mathematics archive .
• Obituary by Fenstad, pdf

Remarks

1. ↑ Albert Thoralf Skolem in the Mathematics Genealogy Project (English)
2. ↑ initially with the axiom of choice , later without its use.
3. ↑ Skolem, Diophantine equations, results of mathematics and their border areas, Springer, Berlin 1938. Represented in Borevich, Shafarevich, number theory, Birkhäuser 1966