# Thoralf Skolem

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

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

## Biography

Skolem was a son of a teacher and studied from 1905 in Kristiania ( Oslo from 1925 called ). From 1909 he worked ( known for his investigations of the northern lights ) for the physicist Kristian Birkeland, with whom he undertook in 1913 an expedition to the Sudan. Skolem dissertation Undersokelser innenfor logikkens algebra ( studies on the algebra of logic ) took a lot of attention and was even reported to the Norwegian king. In 1915 he traveled to Göttingen, where he studied during the winter semester. In 1916 he returned to Christiania and went under Axel Thue a research position at the University, where he initially with there also acting Viggo Brun agreed on, not the doctor's degree to work towards. 1918 Skolem was a lecturer in mathematics in Christiania and in the same year a member of the Norwegian Academy of Sciences.

1926 ranged Skolem a dissertation (Some theorems on integral solutions of certain equations and inequalities ) on number theory one (actually he and his friend Viggo Brun decided to pass up because they did not consider that in Norway necessary ). His real doctor father, the well-known number theorist Axel Thue was, but then died for four years.

1927 married Edith Skolem Wilhelmine Hasvold and continued to work at the University of Oslo, until he and his wife went to Bergen in 1930 to work as a researcher at the Christian Michelsen Institute. He worked there until 1938, when he accepted a call to Oslo, where a chair of mathematics took over, which he held until his retirement in 1957. He held only occasionally lectures on his actual field of mathematical logic and reasoned in Norway also no school. Since he published mostly in Norwegian magazines, some of his results remained unnoticed until others rediscovered. For example, he wrote in 1912 an essay on the theory of organizations and characterized 1927, the automorphisms of simple algebras, which was later re-discovered by Emmy Noether ( Skolem -Noether theorem). Skolem remained scientifically active until his death.

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

## Work

By means of the after named him predicate logic normal form ( Skolem ) he has the Löwenheim (1915 ) that every satisfiable expression of the predicate calculus is satisfiable already in a most countable range in 1920 given a manageable proof, so this set today Löwenheim and Skolem is called. Skolem also dismissed in 1922 on the seemingly paradoxical consequences of this theorem in the axiomatic set theory out ( " Skolem Paradox ").

In 1929 he gave the first precise predicate logic formalization of Zermelo -Fraenkel set theory. By Skolem the final point is set by the comprehension axiom he gave his now customary version with the means of formalization in the axiomatization of set theory. On Skolem the now common notion of primitive recursive function back (1923 ) goes.

He showed that Peano arithmetic is not finite axiomatizable. Skolem also made a number of contributions to the decision problem. From him came the first attempt to build an axiomatic set theory with full comprehension axiom based on a multivalued logic.

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

In the field of algebra, he published in 1927 a day designated as a set of Skolem -Noether theorem, according to which two embeddings of a simple algebra in a central simple algebra differ only by conjugation with an invertible element:

This result thereof was proved independently by Emmy Noether.

## Writings

- Jens Erik Fenstad (Editor) Thoralf Skolem. Selected Works in Logic, Oslo, University Press 1970
- Studies on the axioms calculus of classes and on the Produktations and summation problems which certain classes of statements concern, 1919
- Logical- combinatorial investigations on the feasibility and provable mathematical theorems, together with a theorem on dense volumes, 1920
- Some Remarks on the axiomatic justification of set theory, 1922-1923
- Justification of elementary arithmetic by the recurrent way of thinking without the use of apparent variables with infinite expansion region, 1923
- On the theory of associative Number Systems, 1927
- About some fundamental questions of Mathematics, 1929
- About the basic discussion in mathematics, 1929-1930
- About some set of functions in arithmetic, 1930-1931
- About the impossibility of a complete characterization of the set of numbers by a finite system of axioms, 1933
- About the non- characterizability the number series using finite or countably infinite number of statements with only numerical variables, 1934
- About the satisfiability of certain Zählausdrücke, 1935
- About the reducibility of some defined by recursion relations on ' arithmetic ', 1936-1937
- Sur la Portee de Löwenheim - Skolem, 1938
- Some remarks on the induction schemes in the recursive number theory, 1939
- Some remarks on recursive arithmetic, 1944
- Remarks about the comprehension axiom, 1957