Thoralf Skolem | |
---|---|

Born | |

Died | 23 March 1963 75) Oslo, Norway | (aged

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** (Norwegian: [ˈtùːralf ˈskùːlɛm] ; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory.

Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany.

In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled *Investigations on the Algebra of Logic*. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic, metamathematics, and abstract algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters.

Skolem did not at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled *Some theorems about integral solutions to certain algebraic equations and inequalities*. His notional thesis advisor was Axel Thue, even though Thue had died in 1922.

In 1927, he married Edith Wilhelmine Hasvold.

Skolem continued to teach at Det kongelige Frederiks Universitet (renamed the University of Oslo in 1939) until 1930 when he became a Research Associate in Chr. Michelsen Institute in Bergen. This senior post allowed Skolem to conduct research free of administrative and teaching duties. However, the position also required that he reside in Bergen, a city which then lacked a university and hence had no research library, so that he was unable to keep abreast of the mathematical literature. In 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the graduate courses in algebra and number theory, and only occasionally on mathematical logic. Skolem's Ph.D. student Øystein Ore went on to a career in the USA.

Skolem served as president of the Norwegian Mathematical Society, and edited the *Norsk Matematisk Tidsskrift* ("The Norwegian Mathematical Journal") for many years. He was also the founding editor of *Mathematica Scandinavica*.

After his 1957 retirement, he made several trips to the United States, speaking and teaching at universities there. He remained intellectually active until his sudden and unexpected death.

For more on Skolem's academic life, see Fenstad (1970).

Skolem published around 180 papers on Diophantine equations, group theory, lattice theory, and most of all, set theory and mathematical logic. He mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras. Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later.

Skolem was among the first to write on lattices. In 1912, he was the first to describe a free distributive lattice generated by *n* elements. In 1919, he showed that every implicative lattice (now also called a Skolem lattice) is distributive and, as a partial converse, that every finite distributive lattice is implicative. After these results were rediscovered by others, Skolem published a 1936 paper in German, "Über gewisse 'Verbände' oder 'Lattices'", surveying his earlier work in lattice theory.

Skolem was a pioneer model theorist. In 1920, he greatly simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim–Skolem theorem, which states that if a countable first-order theory has an infinite model, then it has a countable model. His 1920 proof employed the axiom of choice, but he later (1922 and 1928) gave proofs using Kőnig's lemma in place of that axiom. It is notable that Skolem, like Löwenheim, wrote on mathematical logic and set theory employing the notation of his fellow pioneering model theorists Charles Sanders Peirce and Ernst Schröder, including ∏, ∑ as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica, and * Principles of Mathematical Logic *. Skolem (1934) pioneered the construction of non-standard models of arithmetic and set theory.

Skolem (1922) refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic. The resulting axiom is now part of the standard axioms of set theory. Skolem also pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets.

The completeness of first-order logic is a corollary of results Skolem proved in the early 1920s and discussed in Skolem (1928), but he failed to note this fact, perhaps because mathematicians and logicians did not become fully aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's * Principles of Mathematical Logic * clearly articulated it. In any event, Kurt Gödel first proved this completeness in 1930.

Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics. Skolem (1923) sets out his primitive recursive arithmetic, a very early contribution to the theory of computable functions, as a means of avoiding the so-called paradoxes of the infinite. Here he developed the arithmetic of the natural numbers by first defining objects by primitive recursion, then devising another system to prove properties of the objects defined by the first system. These two systems enabled him to define prime numbers and to set out a considerable amount of number theory. If the first of these systems can be considered as a programming language for defining objects, and the second as a programming logic for proving properties about the objects, Skolem can be seen as an unwitting pioneer of theoretical computer science.

In 1929, Presburger proved that Peano arithmetic without multiplication was consistent, complete, and decidable. The following year, Skolem proved that the same was true of Peano arithmetic without addition, a system named Skolem arithmetic in his honor. Gödel's famous 1931 result is that Peano arithmetic itself (with both addition and multiplication) is incompletable and hence * a posteriori * undecidable.

Hao Wang praised Skolem's work as follows:

"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." (Skolem 1970: 17-18)

For more on Skolem's accomplishments, see Hao Wang (1970).

An **axiom** or **postulate** is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek *axíōma* (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'

**Automated theorem proving** is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science.

**Mathematical logic** is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In mathematics, **model theory** is the study of classes of mathematical structures from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is one of the components that form a **theory**. A **model** of a theory is a structure that satisfies the sentences of that theory.

In mathematical logic, the **Peano axioms**, also known as the **Dedekind–Peano axioms** or the **Peano postulates**, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

In mathematical logic, the **compactness theorem** states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.

In set theory, **Zermelo–Fraenkel set theory**, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated **ZFC**, where C stands for "choice", and **ZF** refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

In mathematics, **Hilbert's second problem** was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in Hilbert (1900), which include a second order completeness axiom.

**Ernst Friedrich Ferdinand Zermelo** was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.

**Proof theory** is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

**George Stephen Boolos** was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.

In mathematical logic, the **Löwenheim–Skolem theorem** is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem.

In mathematics, **Hilbert's program**, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.

In mathematical logic and philosophy, **Skolem's paradox** is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox, and was described as a "paradoxical state of affairs" by Skolem.

In mathematics, **Robinson arithmetic**, or **Q**, is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out in R. M. Robinson (1950). **Q** is almost PA without the axiom schema of induction. **Q** is weaker than PA but it has the same language, and both theories are incomplete. **Q** is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.

In mathematical logic, a **non-standard model of arithmetic** is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term **standard model of arithmetic** refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

**Logic** is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

A timeline of **mathematical logic**. See also History of logic.

- Skolem, Thoralf (1934). "Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschließlich Zahlenvariablen" (PDF).
*Fundamenta Mathematicae*(in German).**23**(1): 150–161. - Skolem, T. A., 1970.
*Selected works in logic*, Fenstad, J. E., ed. Oslo: Scandinavian University Books. Contains 22 articles in German, 26 in English, 2 in French, 1 English translation of an article originally published in Norwegian, and a complete bibliography.

- Jean van Heijenoort, 1967.
*From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931*. Harvard Univ. Press.- 1920. "Logico-combinatorial investigations on the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by Löwenheim," 252–263.
- 1922. "Some remarks on axiomatized set theory," 290-301.
- 1923. "The foundations of elementary arithmetic," 302-33.
- 1928. "On mathematical logic," 508–524.

- Brady, Geraldine, 2000.
*From Peirce to Skolem*. North Holland. - Fenstad, Jens Erik, 1970, "Thoralf Albert Skolem in Memoriam" in Skolem (1970: 9–16).
- Hao Wang, 1970, "A survey of Skolem's work in logic" in Skolem (1970: 17–52).

- O'Connor, John J.; Robertson, Edmund F., "Thoralf Skolem",
*MacTutor History of Mathematics archive*, University of St Andrews . - Thoralf Skolem at the Mathematics Genealogy Project
- Fenstad, Jens Erik, 1996, "Thoralf Albert Skolem 1887-1963: A Biographical Sketch,"
*Nordic Journal of Philosophical Logic 1*: 99-106.

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