Thoralf Skolem

Thoralf Albert Skolem was a Norwegian mathematician born on May 23, 1887, and passed away on March 23, 1963. He made important contributions to the fields of mathematical logic and set theory.

Skolem is best known for his work in mathematical logic, where he developed what are now called Skolem functions. These functions help solve certain problems in logic and are important for understanding the foundations of mathematics.

In set theory, Skolem introduced ideas that challenged how mathematicians thought about infinite sets. His work showed that certain assumptions about sets could lead to surprising results, influencing how mathematicians study infinity today.

Because of his important discoveries, Thoralf Skolem is remembered as a key figure in modern mathematics. His ideas continue to be studied and used by mathematicians around the world.

Life

Thoralf Skolem grew up in a family where his father was a teacher and most relatives were farmers. He went to secondary school in Kristiania, now called Oslo, and passed his university exams in 1905. He studied mathematics at Det Kongelige Frederiks Universitet, also taking classes in physics, chemistry, zoology, and botany.

Skolem started working with physicist Kristian Birkeland, and they wrote some physics papers together. In 1913, he finished his dissertation and later earned his Ph.D. in 1926. He taught at the university, which became the University of Oslo in 1939, and later worked at the Chr. Michelsen Institute in Bergen. He returned to Oslo in 1938 as a professor. After retiring in 1957, he visited and taught in the United States until his death.

Mathematics

Thoralf Skolem was a mathematician who wrote many important papers about different areas of math, such as solving equations, groups, and set theory. One of his famous works is called the Skolem–Noether theorem, which helps us understand certain types of algebraic structures. He also made big contributions to a branch of math called lattice theory.

Skolem was one of the first people to study something called model theory. He made simpler proofs for important math theorems and helped create new ways to understand numbers and sets. His ideas are still used by mathematicians today.

Completeness

Thoralf Skolem made important discoveries in mathematical logic. In the early 1920s, he proved results that later helped show the completeness of first-order logic, although he did not realize this at the time. Later, Kurt Gödel formally proved this completeness in 1930.

Skolem was careful about using infinite numbers and helped start a way of thinking in math called finitism. He created a system for studying numbers using primitive recursion, which was an early step toward what we now call computer science. In 1929, another mathematician named Presburger showed that a certain number system without multiplication was consistent, complete, and decidable. The next year, Skolem proved the same for a system without addition, now called Skolem arithmetic. Later, Gödel showed that a fuller number system with both addition and multiplication cannot be complete.