Thoralf Skolem

1.1. Early Life and Family Background

Skolem was born on May 23, 1887. His father was a primary school teacher, though most of his extended family were farmers, suggesting a background rooted in rural Norway. He attended secondary school in Kristiania (which was later renamed Oslo), successfully passing his university entrance examinations in 1905.

1.2. Education

Following his secondary education, Skolem enrolled at Det Kongelige Frederiks Universitet (The Royal Frederick University), which would later become the University of Oslo. There, he pursued a comprehensive curriculum, primarily studying mathematics but also taking courses in diverse scientific fields such as physics, chemistry, zoology, and botany.

1.3. Early Career and Collaboration with Birkeland

In 1909, Skolem began working as an assistant to the renowned physicist Kristian Birkeland. Birkeland was famous for his experiments involving bombarding magnetized spheres with electrons to simulate aurora-like effects. This collaboration led to Skolem's initial publications being physics papers co-authored with Birkeland. Their work together also involved an expedition to Sudan in 1913, where they observed the zodiacal light. That same year, Skolem passed the state examinations with distinction and completed a dissertation titled Investigations on the Algebra of Logic.

1.4. Academic Career and University Life

Skolem's academic career progressed steadily. In 1915, he spent the winter semester at the University of Göttingen, which was then a leading center for research in mathematical logic, metamathematics, and abstract algebra-fields in which Skolem would later excel. The following year, in 1916, he was appointed a research fellow at Det Kongelige Frederiks Universitet. By 1918, he had become a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters.

Initially, Skolem did not formally enroll as a Ph.D. candidate, believing that a doctoral degree was not necessary in Norway. However, he later changed his mind and submitted his thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and inequalities. His notional thesis advisor was Axel Thue, who had unfortunately passed away in 1922.

In 1930, Skolem moved to Bergen to become a Research Associate at the Chr. Michelsen Institute (Christian Michelsens Institutt). This senior position offered him the freedom to conduct research without the burdens of administrative and teaching duties. However, the requirement to reside in Bergen, a city that at the time lacked a university and a comprehensive research library, made it difficult for him to stay updated with the latest mathematical literature. Consequently, in 1938, he returned to Oslo to assume the Professorship of Mathematics at the University of Oslo (renamed from Det Kongelige Frederiks Universitet in 1939). There, he primarily taught graduate courses in algebra and number theory, with mathematical logic being an occasional subject. Throughout his career, Skolem had only one Ph.D. student, Øystein Ore, who later pursued a successful academic career in the United States.

Skolem also played a significant role in the Norwegian mathematical community, serving as president of the Norwegian Mathematical Society and editing the Norsk Matematisk Tidsskrift ("The Norwegian Mathematical Journal") for many years. He was also the founding editor of Mathematica Scandinavica.

1.5. Personal Life

In 1927, Thoralf Skolem married Edith Wilhelmine Hasvold (Edith Wilhelmine Hasvold^Norwegian).

1.6. Retirement and Later Years

After his retirement in 1957, Skolem remained intellectually active. He undertook several trips to the United States, where he lectured and taught at various universities. He continued his academic pursuits until his sudden and unexpected death on March 23, 1963.

2.1. Mathematical Logic and Set Theory

Skolem made foundational contributions to mathematical logic and set theory. In 1922, he refined Zermelo's axioms for set theory. He replaced Zermelo's somewhat vague concept of a "definite" property with any property that could be precisely defined or "coded" within first-order logic. This refined axiom is now an integral part of the standard axioms of set theory.

A significant consequence of the Löwenheim-Skolem theorem, which Skolem helped prove, is what is now known as Skolem's paradox. This paradox states that if Zermelo's axioms for set theory are consistent, then they must be satisfiable within a countable domain, even though these very axioms prove the existence of uncountable sets. This highlights a peculiar aspect of formal systems and their models.

2.2. Model Theory

Skolem was a pioneer in model theory, a branch of mathematical logic that studies the relationships between formal theories and their models. In 1920, he significantly simplified the proof of a theorem that Leopold Löwenheim had first established in 1915. This collaborative effort led to the widely recognized Löwenheim-Skolem theorem, which asserts that if a countable first-order theory possesses an infinite model, then it must also have a countable model. Skolem's 1920 proof initially utilized the axiom of choice, but he later provided alternative proofs in 1922 and 1928, employing Kőnig's lemma instead.

Notably, Skolem, like Löwenheim, adopted the notation of pioneering model theorists Charles Sanders Peirce and Ernst Schröder in his writings on mathematical logic and set theory. This included the use of Π and Σ as variable-binding quantifiers, a departure from the notations used in Peano arithmetic, Principia Mathematica, and Principles of Mathematical Logic. Furthermore, Skolem's work in 1934 was instrumental in pioneering the construction of non-standard models of arithmetic and set theory, which are crucial for understanding the limitations of formal systems.

2.3. Lattice Theory

Skolem was among the earliest mathematicians to conduct research on lattice theory. In 1912, he was the first to describe a free distributive lattice generated by n elements. His 1919 work demonstrated that every implicative lattice (now also referred to as a Skolem lattice) is distributive. As a partial converse, he also showed that every finite distributive lattice is implicative. These results were later rediscovered by others. To survey his earlier work in lattice theory, Skolem published a paper in German in 1936, titled "Über gewisse 'Verbände' oder 'Lattices'".

2.4. Finitism and Computability Theory

Skolem held a philosophical distrust of the completed infinite and is considered one of the founders of finitism in mathematics. Finitism advocates that only finite mathematical objects exist or can be meaningfully used. In his 1923 work, Skolem introduced his primitive recursive arithmetic, an early and significant contribution to the theory of computable functions. He developed this system as a means to avoid the so-called paradoxes associated with the infinite. In this framework, he constructed the arithmetic of the natural numbers by first defining objects through primitive recursion, and then devising a separate system to prove properties of these defined objects. These two systems allowed him to define prime numbers and establish a considerable portion of number theory. This approach has led some to view Skolem as an unwitting pioneer of theoretical computer science, with the first system resembling a programming language for object definition and the second a programming logic for proving properties.

2.5. Key Theorems and Concepts

Skolem's name is associated with several fundamental theorems and concepts in mathematics and logic:

• Skolem-Noether theorem: This theorem characterizes the automorphisms of simple algebras. Skolem published a proof in 1927, though Emmy Noether independently rediscovered it a few years later.
• Löwenheim-Skolem theorem: A cornerstone of model theory, stating that if a countable first-order theory has an infinite model, it also has a countable model.
• Skolem normal form: A concept in mathematical logic related to the transformation of logical formulas into a standardized form.
• Skolem's paradox: The counter-intuitive consequence of the Löwenheim-Skolem theorem applied to set theory, suggesting that consistent axiomatic set theories can have countable models despite proving the existence of uncountable sets.
• Primitive recursive arithmetic: An early formal system for arithmetic based on primitive recursive functions, reflecting Skolem's finitist philosophy.
• Skolem arithmetic: A system of arithmetic similar to Peano arithmetic but without addition. Skolem proved in 1930 that this system is consistent, complete, and decidable, contrasting with Gödel's later 1931 result showing that full Peano arithmetic (with both addition and multiplication) is incompletable and undecidable.
• Skolem-Mahler-Lech theorem: A theorem concerning the zeros of a linear recurrence sequence.
• P-adic numbers: Skolem also made contributions related to P-adic numbers, which are a system of numbers constructed from the rational numbers using a different notion of "closeness" or absolute value.

4.1. Scholarly Assessment

Scholars and peers have consistently praised Skolem's originality and innovative approach. Hao Wang, a prominent logician and philosopher, offered a notable assessment of Skolem's work:
:Skolem tends to treat general problems by concrete examples. 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.
This assessment highlights Skolem's unique style, characterized by concrete examples, informal yet insightful proofs, and a spirit of independent innovation. His ideas were often seen as highly fertile, capable of broad application despite their initial presentation as "progress reports."

4.2. Criticisms and Debates

One notable aspect of Skolem's career was that many of his important results were initially published in Norwegian journals, which had limited international circulation. This sometimes led to his findings being independently rediscovered by other mathematicians who were unaware of his prior work, such as the Skolem-Noether theorem. This situation, while not a criticism of his mathematical rigor, did impact the immediate recognition of his contributions.

Another point of discussion concerns the completeness of first-order logic. Skolem proved results in the early 1920s that were corollaries of this completeness, and he discussed them in his 1928 paper. However, he did not explicitly note the completeness itself. This might have been because the concept of completeness as a fundamental metamathematical problem was not fully articulated until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic. It was Kurt Gödel who first explicitly proved this completeness in 1930.

4.3. Legacy and Impact

Skolem's mathematical ideas and methodologies have had a lasting impact on subsequent developments in logic and mathematics. His pioneering work in model theory, particularly the Löwenheim-Skolem theorem and the construction of non-standard models, laid crucial groundwork for this field. His contributions to lattice theory were foundational, and his development of primitive recursive arithmetic marked an early and significant step in computability theory, influencing later work on recursive functions and the theoretical underpinnings of computer science. Furthermore, his consistent advocacy for finitism stimulated important philosophical debates about the foundations of mathematics and the nature of infinity. Despite his modest publication venues, the depth and originality of Skolem's insights ensured his place as a pivotal figure in 20th-century mathematics.

[https://mathshistory.st-andrews.ac.uk/Biographies/Skolem/ Thoralf Albert Skolem at the MacTutor History of Mathematics Archive]
[https://mathgenealogy.org/id.php?id=18237 Thoralf Albert Skolem at the Mathematics Genealogy Project]