Thoralf Skolem
Encyclopedia
Thoralf Albert Skolem (ˈtuːɾɑlf ˈskuːləm) was a Norwegian
mathematician
known mainly for his work on mathematical logic
and set theory
.
), 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 electron
s 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. Over the course of his entire career, he had but one Ph.D. student, but that student was a productive one, Øystein Ore
, who 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 life, see Fenstad (1970).
, 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 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 König'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.
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 function
s, 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 number
s 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 fortiori undecidable.
Hao Wang praised Skolem's work as follows:
For more on Skolem's accomplishments, see Hao Wang (1970).
Norway
Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known mainly for his work on mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and set theory
Set theory
Set theory is the branch of mathematics that studies sets, which 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...
.
Life
Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed OsloOslo
Oslo is a municipality, as well as the capital and most populous city in Norway. As a municipality , it was established on 1 January 1838. Founded around 1048 by King Harald III of Norway, the city was largely destroyed by fire in 1624. The city was moved under the reign of Denmark–Norway's King...
), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
, zoology
Zoology
Zoology |zoölogy]]), is the branch of biology that relates to the animal kingdom, including the structure, embryology, evolution, classification, habits, and distribution of all animals, both living and extinct...
and botany
Botany
Botany, plant science, or plant biology is a branch of biology that involves the scientific study of plant life. Traditionally, botany also included the study of fungi, algae and viruses...
.
In 1909, he began working as an assistant to the physicist Kristian Birkeland
Kristian Birkeland
Kristian Olaf Birkeland was a Norwegian scientist. He is best remembered as the person who first 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...
, known for bombarding magnetized spheres with electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s and obtaining aurora
Aurora (astronomy)
An aurora is a natural light display in the sky particularly in the high latitude regions, caused by the collision of energetic charged particles with atoms in the high altitude atmosphere...
-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
Zodiacal light
Zodiacal light is a faint, roughly triangular, whitish glow seen in the night sky which appears to extend up from the vicinity of the sun along the ecliptic or zodiac. Caused by sunlight scattered by space dust in the zodiacal cloud, it is so faint that either moonlight or light pollution renders...
. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, metamathematics
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
, and abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, 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
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters is a learned society based in Oslo, Norway.-History:The University of Oslo was established in 1811. The idea of a learned society in Christiania surfaced for the first time in 1841. The city of Throndhjem had no university, but had a learned...
.
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
Axel Thue
Axel Thue was a Norwegian mathematician, known for highly original work in diophantine approximation, and combinatorics....
, 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
University of Oslo
The University of Oslo , formerly The Royal Frederick University , is the oldest and largest university in Norway, situated in the Norwegian capital of Oslo. The university was founded in 1811 and was modelled after the recently established University of Berlin...
in 1939) until 1930 when he became a Research Associate in Chr. Michelsen Institute
Chr. Michelsen Institute
The Chr. Michelsen Institute was founded in 1930, and is currently the largest centre for development research in Scandinavia. 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...
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. Over the course of his entire career, he had but one Ph.D. student, but that student was a productive one, Øystein Ore
Øystein Ore
Øystein Ore was a Norwegian mathematician.-Life:Ore was graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem...
, who 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 life, see Fenstad (1970).
Mathematics
Skolem published around 180 papers on Diophantine equations, group theoryGroup theory
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...
, lattice theory, and most of all, set theory
Set theory
Set theory is the branch of mathematics that studies sets, which 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...
and mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
. 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
Skolem–Noether theorem
In mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in ring theory which characterizes the automorphisms of simple rings....
, characterizing the automorphisms of simple algebras. Skolem published a proof in 1927, but Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
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
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
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
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
. In 1920, he greatly simplified the proof of a theorem Leopold Löwenheim
Leopold Löwenheim
Leopold Löwenheim was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin...
first proved in 1915, resulting in the Löwenheim–Skolem theorem
Löwenheim–Skolem theorem
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ...
, which states that if a 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 König's lemma
König's lemma
König's lemma or König's infinity lemma is a theorem in graph theory due to Dénes Kőnig . It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic,...
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
Ernst Schröder
Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
, including ∏, ∑ as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
, 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
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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
Skolem's paradox
In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as...
: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets.
Completeness
The completenessGödel's completeness theorem
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 was first proved by Kurt Gödel in 1929....
of first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
first proved this completeness in 1930.
Skolem distrusted the completed infinite and was one of the founders of finitism
Finitism
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
in mathematics. Skolem (1923) sets out his primitive recursive arithmetic
Primitive recursive arithmetic
Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers. It was first proposed by Skolem as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist...
, a very early contribution to the theory of computable function
Computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register...
s, 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 number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s 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
Presburger arithmetic
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely...
proved that Peano arithmetic without multiplication was consistent, complete, and decidable
Decidable
The word decidable may refer to:* Decidable language*Decidability for the equivalent in mathematical logic*Gödel's incompleteness theorem, a theorem on the indecidability of languages consisting of "true statements" in mathematical logic....
. The following year, Skolem proved that the same was true of Peano arithmetic without addition, a system named Skolem arithmetic in his honor. Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
's famous 1931 result is that Peano arithmetic itself (with both addition and multiplication) is incompletable
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
and hence a fortiori 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).
See also
- Leopold LöwenheimLeopold LöwenheimLeopold Löwenheim was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin...
- Model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
- Skolem normal formSkolem normal formReduction to Skolem normal form is a method for removing existential quantifiers from formal logic statements, often performed as the first step in an automated theorem prover....
- Skolem's paradoxSkolem's paradoxIn mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as...
- Skolem sequence
External links
- MacTutor: Thoralf Skolem.
- Fenstad, Jens Erik, 1996, "Thoralf Albert Skolem 1887-1963: A Biographical Sketch," Nordic Journal of Philosophical Logic 1: 99-106.