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

Born | |

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

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

**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.

**Set theory** is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.

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.

**Oslo** is the capital and most populous city of Norway. It constitutes both a county and a municipality. Founded in the year 1040 as **Ánslo**, and established as a *kaupstad* or trading place in 1048 by Harald Hardrada, the city was elevated to a bishopric in 1070 and a capital under Haakon V of Norway around 1300. Personal unions with Denmark from 1397 to 1523 and again from 1536 to 1814 reduced its influence. After being destroyed by a fire in 1624, during the reign of King Christian IV, a new city was built closer to Akershus Fortress and named **Christiania** in the king's honour. It was established as a municipality (*formannskapsdistrikt*) on 1 January 1838. The city functioned as a co-official capital during the 1814 to 1905 Union between Sweden and Norway. In 1877, the city's name was respelled **Kristiania** in accordance with an official spelling reform – a change that was taken over by the municipal authorities only in 1897. In 1925 the city, after incorporating the village retaining its former name, was renamed Oslo.

**Physics** is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

**Chemistry** is the scientific discipline involved with elements and compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances.

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.

**Kristian Olaf Bernhard Birkeland** was a Norwegian scientist. He is best remembered for his theories of atmospheric electric currents that elucidated the nature of the aurora borealis. In order to fund his research on the aurorae, he invented the electromagnetic cannon and the Birkeland-Eyde process of fixing nitrogen from the air. Birkeland was nominated for the Nobel Prize seven times.

The **electron** is a subatomic particle, symbol ^{}e^{−}_{} or ^{}β^{−}_{}, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, *ħ*. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

**Zodiacal light** is a faint, diffuse, and roughly triangular white glow that is visible in the night sky and appears to extend from the Sun's direction and along the zodiac, straddling the ecliptic. Sunlight scattered by interplanetary dust causes this phenomenon. Zodiacal light is best seen during twilight after sunset in spring and before sunrise in autumn, when the zodiac is at a steep angle to the horizon. However, the glow is so faint that moonlight and/or light pollution outshine it, rendering it invisible.

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.

**Axel Thue**, was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics.

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.

The **University of Oslo**, until 1939 named the **Royal Frederick University**, is the oldest university in Norway, located in the Norwegian capital of Oslo. Until 1 January 2016 it was the largest Norwegian institution of higher education in terms of size, now surpassed only by the Norwegian University of Science and Technology. The Academic Ranking of World Universities has ranked it the 58th best university in the world and the third best in the Nordic countries. In 2015, the Times Higher Education World University Rankings ranked it the 135th best university in the world and the seventh best in the Nordics. While in its 2016, Top 200 Rankings of European universities, the Times Higher Education listed the University of Oslo at 63rd, making it the highest ranked Norwegian university.

The **Chr. Michelsens Institutt for Videnskap og Åndsfrihet** (CMI) was founded in 1930, and is currently the largest centre for development research in Scandinavia. In 1992, the Department for Natural Science and Technology established the *Christian Michelsen Research AS*, and the CMR Group.(cmr.no) The University of Bergen is the main owner in The CMR Group. The Department of Social Science and Development became the Chr. Michelsen Institute. CMI is an independent, non-profit research foundation for policy-oriented and applied development research. Headed by the director Ottar Mæstad, it employs 40 social scientists, primarily anthropologists, economists and political scientists. CMI receives core funding from the Norwegian Research Council (NFR) and project support from Norwegian state ministries and agencies, and Norwegian and international non-governmental organisations.

**Øystein Ore** was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics.

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.

In mathematics and abstract algebra, **group theory** studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In ring theory, a branch of mathematics, the **Skolem–Noether theorem** characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras.

**Amalie Emmy Noether** was a German mathematician who made important contributions to abstract algebra and theoretical physics. She invariably used the name "Emmy Noether" in her life and publications. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed the theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

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 an easy 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).

**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.

**Gödel's completeness theorem** is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.

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.

**Gödel's incompleteness theorems** are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

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.

**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.

**Reverse mathematics** is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.

In mathematical logic, the **Löwenheim–Skolem theorem** is a theorem named after Leopold Löwenheim and Thoralf Skolem.

In mathematics, the **axiom of dependent choice**, denoted by , is a weak form of the axiom of choice that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.

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 mathematical logic, a **conservative extension** is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a **non-conservative extension** is a supertheory which is not conservative, and can prove more theorems than the original.

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.

- Thoralf Skolem (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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