Thoralf Skolem facts for kids

Quick facts for kids Thoralf Skolem
Born	(1887-05-23)23 May 1887 Sandsvær, Norway
Died	23 March 1963(1963-03-23) (aged 75) Oslo, Norway
Nationality	Norwegian
Alma mater	Oslo University
Known for	Skolem–Noether theorem Löwenheim–Skolem theorem
Scientific career
Fields	Mathematician
Institutions	Oslo University Chr. Michelsen Institute
Doctoral advisor	Axel Thue
Doctoral students	Øystein Ore

Thoralf Albert Skolem (born May 23, 1887 – died March 23, 1963) was a famous Norwegian mathematician. He worked on mathematical logic and set theory. These are branches of mathematics that deal with the rules of reasoning and collections of objects.

A Look at Skolem's Life

Thoralf Skolem grew up in Norway. His father was a teacher, but many of his relatives were farmers. He went to secondary school in Kristiania, which is now called Oslo. In 1905, he passed his university entrance exams. He then started studying mathematics at the Royal Frederick University. He also took classes in physics, chemistry, and biology.

Early Career and Studies

In 1909, Skolem began working with a physicist named Kristian Birkeland. Birkeland was known for his work with electrons and auroras. Skolem's first published papers were actually about physics, written with Birkeland.

Skolem finished his main studies in 1913. He wrote a paper called Investigations on the Algebra of Logic. He also traveled to Sudan with Birkeland to study the zodiacal light. This is a faint glow seen in the night sky.

In 1915, Skolem spent a semester at the University of Göttingen in Germany. This university was a top place for studying mathematical logic. In 1916, he became a research fellow at his university in Norway. He was later made a mathematics teacher in 1918. He also joined the Norwegian Academy of Science and Letters.

Becoming a Professor

Skolem didn't get a Ph.D. degree at first. He thought it wasn't needed in Norway. But he later changed his mind. In 1926, he submitted a thesis. A thesis is a long paper written for a university degree. His paper was about solving certain math equations.

In 1927, he married Edith Wilhelmine Hasvold.

Skolem taught at the University of Oslo until 1930. Then he became a researcher at the Chr. Michelsen Institute in Bergen. This job let him focus on research without teaching. However, Bergen didn't have a university library at the time. This made it hard for him to keep up with new math discoveries.

In 1938, he returned to Oslo. He became a Professor of Mathematics at the university. He mostly taught advanced courses in algebra and number theory. His student, Øystein Ore, later became a successful mathematician in the USA.

Skolem was also involved in math organizations. He was president of the Norwegian Mathematical Society. He also edited the Norwegian Mathematical Journal for many years. He helped start another math journal called Mathematica Scandinavica.

After he retired in 1957, Skolem visited the United States many times. He gave talks and taught at universities there. He kept working on math until he passed away suddenly in 1963.

Skolem's Work in Mathematics

Skolem wrote about 180 papers on different math topics. These included Diophantine equations (equations with only whole number solutions), group theory (the study of symmetry), and lattice theory (the study of ordered structures). Most of his work was in set theory and mathematical logic.

He often published his work in Norwegian journals. These journals were not widely known around the world. Because of this, some of his discoveries were later made again by other mathematicians.

Important Theorems

One example is the Skolem–Noether theorem. This theorem describes how certain math structures behave. Skolem proved it in 1927. But another mathematician, Emmy Noether, found the same proof a few years later.

Skolem was also one of the first to write about lattices. In 1912, he described a special type of lattice. In 1919, he showed that certain lattices are always "distributive." This means they follow a specific rule for how their parts combine.

Skolem was a pioneer in model theory. This field studies how mathematical ideas can be represented in different ways. In 1920, he helped simplify the proof of the Löwenheim–Skolem theorem. This theorem says that if a set of math rules has an infinite model, it can also have a smaller, countable model. A "countable" model means you could list its parts, even if there are infinitely many.

Set Theory and Paradoxes

In 1922, Skolem improved the rules for set theory. Set theory is about collections of objects. He made the rules more precise. These rules are now a standard part of set theory.

Skolem also pointed out something called Skolem's paradox. The Löwenheim–Skolem theorem suggests that if set theory rules are consistent, they can be true in a countable world. But set theory also proves that there are "uncountable" sets, which are much larger than countable ones. This seems like a paradox!

Logic and Computable Functions

Skolem's work in the early 1920s helped show that first-order logic is "complete." This means that if a statement in this logic is true, it can be proven. However, Kurt Gödel was the first to formally prove this in 1930.

Skolem was careful about the idea of a "completed infinite." He was one of the founders of finitism. This idea suggests that we should only use finite (limited) steps in mathematics.

In 1923, Skolem developed something called primitive recursive arithmetic. This was an early step towards understanding computable functions. These are functions that can be calculated by a step-by-step process. He created a system to define numbers and prove things about them. This work was like an early form of computer science.

In 1929, another mathematician proved that a simple version of arithmetic (without multiplication) was consistent and complete. The next year, Skolem showed the same was true for arithmetic without addition. This system is now called Skolem arithmetic. Later, Gödel showed that full arithmetic (with both addition and multiplication) cannot be fully proven or decided.

Hao Wang, another mathematician, praised Skolem's work. He said Skolem often explained complex problems using simple examples. He also noted that Skolem was a "free spirit" who didn't follow any specific school of thought. His ideas were often very important and could be used in many ways. Wang said that Skolem's papers are easy to read and understand, even for those without a lot of special knowledge.

See also

• Leopold Löwenheim
• Model theory
• Skolem arithmetic
• Skolem normal form
• Skolem's paradox
• Skolem problem
• Skolem sequence
• Skolem–Mahler–Lech theorem