Thoralf Albert Skolem ( Norwegian: [ˈtûːrɑɫf ˈskûːlɛm] ; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory .
128-469: 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
256-489: A 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} to b 1 , … , b n {\displaystyle b_{1},\dots ,b_{n}} respectively, then a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} and b 1 , … , b n {\displaystyle b_{1},\dots ,b_{n}} realise
384-416: A 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} . This is called the complete (n-)type realised by a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} over A . If there is an automorphism of M {\displaystyle {\mathcal {M}}} that is constant on A and sends
512-480: A 2 {\displaystyle a_{1}=a_{2}} or a 2 < a 1 {\displaystyle a_{2}<a_{1}} . Over the subset Z ⊆ R {\displaystyle \mathbb {Z} \subseteq \mathbb {R} } of integers, the 1-type of a non-integer real number a depends on its value rounded down to the nearest integer. More generally, whenever M {\displaystyle {\mathcal {M}}}
640-480: A n {\displaystyle a_{1},\dots ,a_{n}} of a structure M {\displaystyle {\mathcal {M}}} and a subset A of M {\displaystyle {\mathcal {M}}} , one can consider the set of all first-order formulas φ ( x 1 , … , x n ) {\displaystyle \varphi (x_{1},\dots ,x_{n})} with parameters in A that are satisfied by
768-408: A ) {\displaystyle \varphi (a)} is true. In this way, one can study definable groups and fields in general structures, for instance, which has been important in geometric stability theory. One can even go one step further, and move beyond immediate substructures. Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of
896-496: 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
1024-406: A and b are connected by the order automorphism that shifts all numbers by b-a . The complete 2-type over the empty set realised by a pair of numbers a 1 , a 2 {\displaystyle a_{1},a_{2}} depends on their order: either a 1 < a 2 {\displaystyle a_{1}<a_{2}} , a 1 =
1152-470: A subarctic climate (Köppen climate classification: Dfc ) in the 1991–2020 base period. and Weather Atlas Oslo has many parks and green areas within the city core, as well as outside it. Oslo (with neighbouring Sandvika-Asker) is built in a horseshoe shape on the shores of the Oslofjord and limited in most directions by hills and forests. As a result, any point within the city is relatively close to
1280-411: A Danish king ( Christian IV of Denmark ) was inappropriate as the name of the capital of their country. The municipality developed new areas such as Ullevål garden city (1918–1926) and Torshov (1917–1925). City Hall was constructed in the former slum area of Vika from 1931 to 1950. In 1948, Oslo merged with Aker , a municipality which surrounded the capital and was 27 times larger, thus creating
1408-523: 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
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#17328699330631536-468: A certain type over A is called type-definable over A . For an algebraic example, suppose M {\displaystyle M} is an algebraically closed field . The theory has quantifier elimination . This allows us to show that a type is determined exactly by the polynomial equations it contains. Thus the set of complete n {\displaystyle n} -types over a subfield A {\displaystyle A} corresponds to
1664-463: A city at the end of the Viking Age in 1040 under the name 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 the year 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
1792-399: A curve. In general, definable sets without quantifiers are easy to describe, while definable sets involving possibly nested quantifiers can be much more complicated. This makes quantifier elimination a crucial tool for analysing definable sets: A theory T has quantifier elimination if every first-order formula φ( x 1 , ..., x n ) over its signature is equivalent modulo T to
1920-470: A few years, new large scale housing areas and infrastructure projects are being built and planned across the city, notably in Hasle , Helsfyr , Bjørvika , Nydalen and Sinsen , this is increasing the density in and around Ring 2 and Ring 3 . The Fornebu line on the metro is under construction as of 2024, and further subway lines crossing the inner city are being planned. Oslo occupies an arc of land at
2048-451: 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. The old site east of the Aker river was not abandoned, however, and the village of Oslo remained as a suburb outside the city gates. The suburb called Oslo was eventually included in the city proper. In 1925 the name of the suburb was transferred to
2176-510: A first-order formula ψ( x 1 , ..., x n ) without quantifiers, i.e. ∀ x 1 … ∀ x n ( ϕ ( x 1 , … , x n ) ↔ ψ ( x 1 , … , x n ) ) {\displaystyle \forall x_{1}\dots \forall x_{n}(\phi (x_{1},\dots ,x_{n})\leftrightarrow \psi (x_{1},\dots ,x_{n}))} holds in all models of T . If
2304-641: A formal coat of arms , but which uses a city seal instead. The seal of Oslo shows the city's patron saint , St. Hallvard , with his attributes, the millstone and arrows , with a naked woman at his feet. He is seated on a throne with lion decorations, which at the time was also commonly used by the Norwegian kings . Oslo has various nicknames and names in other languages. The city is sometimes known as "The Tiger City" ( Norwegian : Tigerstaden ), probably inspired by an 1870 poem by Bjørnstjerne Bjørnson which referenced then-Christiania in central Oslo. The nickname
2432-452: A formula of the following form: where ψ is quantifier free. A theory that is not model-complete may have a model completion , which is a related model-complete theory that is not, in general, an extension of the original theory. A more general notion is that of a model companion . In every structure, every finite subset { a 1 , … , a n } {\displaystyle \{a_{1},\dots ,a_{n}\}}
2560-415: 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
2688-443: A model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively: where universal algebra stands for mathematical structures and logic for logical theories; and where logical formulas are to definable sets what equations are to varieties over a field. Nonetheless, the interplay of classes of models and the sets definable in them has been crucial to
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#17328699330632816-484: A model of T which is itself a model of T is an elementary substructure. There is a useful criterion for testing whether a substructure is an elementary substructure, called the Tarski–Vaught test . It follows from this criterion that a theory T is model-complete if and only if every first-order formula φ( x 1 , ..., x n ) over its signature is equivalent modulo T to an existential first-order formula, i.e.
2944-564: A new financial district and a cultural city. In 2008, an exhibition was held in London presenting the award-winning Oslo Opera House , the urban regeneration scheme of Oslo's seafront, Munch/Stenersen and the new Deichman Library. Most of the buildings in the city and in neighbouring communities are low in height with only the Plaza , Posthuset and the highrises at Bjørvika considerably taller. Model theory In mathematical logic , model theory
3072-408: A population of 717,710 as of 1 January 2024. The urban area extends far beyond the boundaries of the municipality into the surrounding county of Akershus (municipalities of Asker , Bærum , Lillestrøm , Enebakk , Rælingen , Lørenskog , Nittedal , Gjerdrum , Nordre Follo ); being, to a great degree suburbs of Oslo making up approximately 500,000 of the population of the greater Oslo region .
3200-444: A preceding urban settlement. This called for the celebration of Oslo's millennium in 2000 rather than 2049. Under the reign of Olaf III of Norway (1067-1093), Oslo became a cultural centre for Eastern Norway . Hallvard Vebjørnsson became the city's patron saint and is depicted on the city's seal. In 1174, Hovedøya Abbey was built. The churches and abbeys became major owners of large tracts of land, which proved important for
3328-477: A river "Lo" predating the work where Peder Claussøn Friis first proposed this etymology, but the very name is ungrammatical in Norwegian: the correct form would have been Loaros (cf. Nidaros ). The name Lo is now believed to be a back-formation arrived at by Friis in support of his [idea about] etymology for Oslo . Oslo is one of very few cities in Norway, besides Bergen and Tønsberg , that does not have
3456-424: A structure are the isolated types, then the structure is called atomic . On the other hand, no structure realises every type over every parameter set; if one takes all of M {\displaystyle {\mathcal {M}}} as the parameter set, then every 1-type over M {\displaystyle {\mathcal {M}}} realised in M {\displaystyle {\mathcal {M}}}
3584-445: A structure is also called the theory of that structure . It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems ) that a theory has a model if and only if it is consistent , i.e. no contradiction is proved by the theory. Therefore, model theorists often use "consistent" as a synonym for "satisfiable". A signature or language is a set of non-logical symbols such that each symbol
3712-404: A theory does not have quantifier elimination, one can add additional symbols to its signature so that it does. Axiomatisability and quantifier elimination results for specific theories, especially in algebra, were among the early landmark results of model theory. But often instead of quantifier elimination a weaker property suffices: A theory T is called model-complete if every substructure of
3840-431: A σ'-theory, and one can extend it (in more than one way) to a complete σ'-theory. The terms reduct and expansion are sometimes applied to this relation as well. The compactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. The analogous statement with consistent instead of satisfiable is trivial, since every proof can have only a finite number of antecedents used in
3968-457: Is Marka (1,610 residents, 301.1 km ), that is administered by several boroughs; and Sentrum (1,471 residents, 1.8 km ) that is partially administered by St. Hanshaugen, and in part directly by the city council. As of 27 February 2020, there were 2,386 residents who were not allocated to a borough. The 9 municipalities directly included in the Urban area of Oslo. After being destroyed by
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4096-472: Is a 1-type over Z ⊆ R {\displaystyle \mathbb {Z} \subseteq \mathbb {R} } that is not realised in the real number line R {\displaystyle \mathbb {R} } . A subset of M n {\displaystyle {\mathcal {M}}^{n}} that can be expressed as exactly those elements of M n {\displaystyle {\mathcal {M}}^{n}} realising
4224-458: Is complete . The set of complete n -types over A is often written as S n M ( A ) {\displaystyle S_{n}^{\mathcal {M}}(A)} . If A is the empty set, then the type space only depends on the theory T {\displaystyle T} of M {\displaystyle {\mathcal {M}}} . The notation S n ( T ) {\displaystyle S_{n}(T)}
4352-452: Is syntactic in nature, in contrast to model theory, which is semantic in nature. The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic . This page focuses on finitary first order model theory of infinite structures. The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within
4480-446: Is July 1901 with mean 22.7 °C (72.9 °F), and the all-time high 35 °C (95 °F) was also recorded in July 1901. The warmest month in more recent years is July 2018 with mean 22.2 °C (72.0 °F) and average daily high 29 °C (84.2 °F). The record summer of 2018 also recorded the warmest May and May all-time high with 31.1 °C (88.0 °F) on 30th, and 2018
4608-559: Is a finite union of points and intervals. Particularly important are those definable sets that are also substructures, i. e. contain all constants and are closed under function application. For instance, one can study the definable subgroups of a certain group. However, there is no need to limit oneself to substructures in the same signature. Since formulas with n free variables define subsets of M n {\displaystyle {\mathcal {M}}^{n}} , n -ary relations can also be definable. Functions are definable if
4736-464: Is a historic timber framing house located on the north side of Christiania Torv; it was built in 1640. The transformation of Christiania went slowly for the first hundred years. However, outside of the city at Vaterland , Grønland , and the old town of Oslo , a new, unmanaged part of the city arose populated by citizens of lower class status. The last Black Death outbreak in Oslo occurred in 1654. In
4864-405: Is a map f : A → B between the domains which can be written as an isomorphism of A {\displaystyle {\mathcal {A}}} with a substructure of B {\displaystyle {\mathcal {B}}} . If it can be written as an isomorphism with an elementary substructure, it is called an elementary embedding. Every embedding is an injective homomorphism, but
4992-408: Is a model of a theory exactly when the superstructure is a model. Example: While the field of algebraic numbers Q ¯ {\displaystyle {\overline {\mathbb {Q} }}} is an elementary substructure of the field of complex numbers C {\displaystyle \mathbb {C} } , the rational field Q {\displaystyle \mathbb {Q} }
5120-456: Is a sentence and A {\displaystyle {\mathcal {A}}} an elementary substructure of B {\displaystyle {\mathcal {B}}} , then A ⊨ φ {\displaystyle {\mathcal {A}}\models \varphi } if and only if B ⊨ φ {\displaystyle {\mathcal {B}}\models \varphi } . Thus, an elementary substructure
5248-432: Is a structure and A a subset of M {\displaystyle {\mathcal {M}}} , a (partial) n-type over A is a set of formulas p with at most n free variables that are realised in an elementary extension N {\displaystyle {\mathcal {N}}} of M {\displaystyle {\mathcal {M}}} . If p contains every such formula or its negation, then p
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5376-404: Is a structure that models T {\displaystyle T} . A substructure A {\displaystyle {\mathcal {A}}} of a σ-structure B {\displaystyle {\mathcal {B}}} is a subset of its domain, closed under all functions in its signature σ, which is regarded as a σ-structure by restricting all functions and relations in σ to
5504-419: Is a subset of Q 2 {\displaystyle \mathbb {Q} ^{2}} ), one obtains a structure ( Q , σ o r ) {\displaystyle (\mathbb {Q} ,\sigma _{or})} . A structure N {\displaystyle {\mathcal {N}}} is said to model a set of first-order sentences T {\displaystyle T} in
5632-535: Is a unary (= 1-ary) function symbol, and < {\displaystyle <} is a binary relation symbol. Then, when these symbols are interpreted to correspond with their usual meaning on Q {\displaystyle \mathbb {Q} } (so that e.g. + {\displaystyle +} is a function from Q 2 {\displaystyle \mathbb {Q} ^{2}} to Q {\displaystyle \mathbb {Q} } and < {\displaystyle <}
5760-520: Is agricultural. The open areas within the built-up zone amount to 22 km (8.5 sq mi). The city of Oslo was established as a municipality on 3 January 1838 (see formannskapsdistrikt ). It was separated from the county of Akershus to become a county of its own in 1842. The rural municipality of Aker was merged with Oslo on 1 January 1948 (and simultaneously transferred from Akershus county to Oslo county). Furthermore, Oslo shares several important functions with Akershus county. In addition
5888-594: Is also a hub of Norwegian trade, banking, industry and shipping. It is an important centre for maritime industries and maritime trade in Europe. The city is home to many companies within the maritime sector, some of which are among the world's largest shipping companies, shipbrokers and maritime insurance brokers. Oslo is a pilot city of the Council of Europe and the European Commission intercultural cities programme. Oslo
6016-513: Is also a main source of drinking water for large parts of Oslo. Although Eastern Norway has a number of rivers, none of these flow into the ocean at Oslo. Instead Oslo has two smaller rivers: Akerselva (draining Maridalsvannet, which flows into the fjord in Bjørvika ), and Alna . The waterfalls in Akerselva gave power to some of the first modern industry of Norway in the 1840s. Later in the century,
6144-486: Is called isolated . Since the real numbers R {\displaystyle \mathbb {R} } are Archimedean , there is no real number larger than every integer. However, a compactness argument shows that there is an elementary extension of the real number line in which there is an element larger than any integer. Therefore, the set of formulas { n < x | n ∈ Z } {\displaystyle \{n<x|n\in \mathbb {Z} \}}
6272-497: Is called a (first-order) theory , which takes the sentences in the set as its axioms. A theory is satisfiable if it has a model M ⊨ T {\displaystyle {\mathcal {M}}\models T} , i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T {\displaystyle T} . A complete theory is a theory that contains every sentence or its negation. The complete theory of all sentences satisfied by
6400-506: Is commonly used for the set of types over the empty set consistent with T {\displaystyle T} . If there is a single formula φ {\displaystyle \varphi } such that the theory of M {\displaystyle {\mathcal {M}}} implies φ → ψ {\displaystyle \varphi \rightarrow \psi } for every formula ψ {\displaystyle \psi } in p , then p
6528-569: Is considered a global city and was ranked "Beta World City" in studies carried out by the Globalization and World Cities Study Group and Network in 2008. It was ranked number one in terms of quality of life among European large cities in the European Cities of the Future 2012 report by fDi magazine. A survey conducted by ECA International in 2011 placed Oslo as the second most expensive city in
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#17328699330636656-439: Is definable by a quantifier-free formula in one variable. Quantifier-free formulas in one variable express Boolean combinations of polynomial equations in one variable, and since a nontrivial polynomial equation in one variable has only a finite number of solutions, the theory of algebraically closed fields is strongly minimal. On the other hand, the field R {\displaystyle \mathbb {R} } of real numbers
6784-552: Is definable with parameters: Simply use the formula Since we can negate this formula, every cofinite subset (which includes all but finitely many elements of the domain) is also always definable. This leads to the concept of a minimal structure . A structure M {\displaystyle {\mathcal {M}}} is called minimal if every subset A ⊆ M {\displaystyle A\subseteq {\mathcal {M}}} definable with parameters from M {\displaystyle {\mathcal {M}}}
6912-451: Is either a constant symbol, or a function or relation symbol with a specified arity . Note that in some literature, constant symbols are considered as function symbols with zero arity, and hence are omitted. A structure is a set M {\displaystyle M} together with interpretations of each of the symbols of the signature as relations and functions on M {\displaystyle M} (not to be confused with
7040-501: Is either finite or cofinite. The corresponding concept at the level of theories is called strong minimality : A theory T is called strongly minimal if every model of T is minimal. A structure is called strongly minimal if the theory of that structure is strongly minimal. Equivalently, a structure is strongly minimal if every elementary extension is minimal. Since the theory of algebraically closed fields has quantifier elimination, every definable subset of an algebraically closed field
7168-430: Is great for barbecues, swimming, beach volleyball and other activities. The municipality operates eight public swimming pools. Tøyenbadet is the largest indoor swimming facility in Oslo and one of the few pools in Norway offering a 50-metre main pool. Another in that size is the outdoor pool Frognerbadet . Oslo's cityscape is being redeveloped as a modern city with various access-points, an extensive metro-system with
7296-419: Is isolated by a formula of the form a = x for an a ∈ M {\displaystyle a\in {\mathcal {M}}} . However, any proper elementary extension of M {\displaystyle {\mathcal {M}}} contains an element that is not in M {\displaystyle {\mathcal {M}}} . Therefore, a weaker notion has been introduced that captures
7424-763: Is mostly used by Norwegians from out of town, and rarely by people from the Oslo region. During the Viking Age , the area that includes modern Oslo was located in Viken , the northernmost province of Denmark . Control over the area shifted between Danish and Norwegian kings in the Middle Ages, and Denmark continued to claim the area until 1241. According to the Norse sagas , Oslo was founded around 1049 by Harald Hardrada . Recent archaeological research, however, uncovered Christian burials which could be dated to prior to AD 1000, evidence of
7552-479: Is not minimal: Consider, for instance, the definable set This defines the subset of non-negative real numbers, which is neither finite nor cofinite. One can in fact use φ {\displaystyle \varphi } to define arbitrary intervals on the real number line. It turns out that these suffice to represent every definable subset of R {\displaystyle \mathbb {R} } . This generalisation of minimality has been very useful in
7680-413: Is not, as we can express "There is a square root of 2" as a first-order sentence satisfied by C {\displaystyle \mathbb {C} } but not by Q {\displaystyle \mathbb {Q} } . An embedding of a σ-structure A {\displaystyle {\mathcal {A}}} into another σ-structure B {\displaystyle {\mathcal {B}}}
7808-454: Is often less concerned with formal rigour and closer in spirit to classical mathematics. This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane" . The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory
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#17328699330637936-573: Is the Löwenheim-Skolem theorem . According to the Löwenheim-Skolem Theorem, every infinite structure in a countable signature has a countable elementary substructure. Conversely, for any infinite cardinal κ every infinite structure in a countable signature that is of cardinality less than κ can be elementarily embedded in another structure of cardinality κ (There is a straightforward generalisation to uncountable signatures). In particular,
8064-462: 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
8192-403: Is the capital and most populous city of Norway . It constitutes both a county and a municipality . The municipality of Oslo had a population of 709,037 in 2022, while the city's greater urban area had a population of 1,064,235 in 2022, and the metropolitan area had an estimated population of 1,546,706 in 2021. During the Viking Age , the area was part of Viken . Oslo was founded as
8320-407: Is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure ), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with
8448-578: Is used, an oceanic climate ( Cfb ) bordering on a humid continental climate in the 1991–2020 base period. Oslo has some of the warmest summers of Norway and fairly cold winters. Oslo receives a fair amount of precipitation during the year. The driest seasons are winter and spring, and the wettest are summer and autumn. Because of the city's northern latitude, daylight varies greatly, from more than 18 hours in midsummer, when it never gets completely dark at night (no darker than nautical twilight ), to around 6 hours in midwinter. The warmest month on record
8576-555: Is −29.6 °C (−21.3 °F), on 21 January 1841, while the coldest recorded at Blindern is −26 °C (−14.8 °F) in January 1941. The coldest temperature more recently was on 6 January 2024, where the temperature reached -23.1 °C (-17 °F) at Oslo-Blindern, which is the coldest measured temperature since January 1987. The coldest month on record is January 1941 and also January 1947 with mean −12.9 °C (8.8 °F) and average daily low −16.7 °C (1.9 °F). The average date for
8704-618: The 2011 Norway attacks , Oslo was hit by a bomb blast that ripped through the Government Quarter , damaging several buildings including the building that houses the Office of the Prime Minister . Eight people died in the bomb attack. On 25 June 2022, two people were killed and 21 others injured in a mass shooting . An Iranian-born Norwegian citizen was subsequently charged with "aggravated terrorism". The city has continued to expand. For
8832-448: The Boolean connectives ¬ , ∧ , ∨ , → {\displaystyle \neg ,\land ,\lor ,\rightarrow } and prefixing of quantifiers ∀ v {\displaystyle \forall v} or ∃ v {\displaystyle \exists v} . A sentence is a formula in which each occurrence of a variable is in
8960-589: 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 Π, Σ as variable-binding quantifiers, in contrast to
9088-609: 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
9216-422: The rational numbers , regarded as a structure in the signature {+,0} can be expanded to a field with the signature {×,+,1,0} or to an ordered group with the signature {+,0,<}. Similarly, if σ' is a signature that extends another signature σ, then a complete σ'-theory can be restricted to σ by intersecting the set of its sentences with the set of σ-formulas. Conversely, a complete σ-theory can be regarded as
9344-595: 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
9472-480: The 18th century, after the Great Northern War , the city's economy boomed with shipbuilding and trade. The strong economy transformed Christiania into a trading port. The Royal Frederick University (now the University of Oslo ) was founded in 1811; the fact that it was founded this late reportedly had an adverse effect on the development of the nation. In 1814 the former provincial town of Christiania became
9600-872: The 19th century, including the Royal Palace (1825-1848), the Stock Exchange (1826-1828), the Bank of Norway (1828), the Storting (1861-1866), the National Theatre (1899), and several University buildings. Among the world-famous artists who lived here during this period were Henrik Ibsen and Knut Hamsun (the latter was awarded the Nobel Prize for literature). Large areas of the surrounding Aker municipality were incorporated in 1839, 1859, and 1878. The 1859 expansion included Grünerløkka , Grønland , and Oslo . At that time
9728-541: The Löwenheim-Skolem Theorem implies that any theory in a countable signature with infinite models has a countable model as well as arbitrarily large models. In a certain sense made precise by Lindström's theorem , first-order logic is the most expressive logic for which both the Löwenheim–Skolem theorem and the compactness theorem hold. In model theory, definable sets are important objects of study. For instance, in N {\displaystyle \mathbb {N} }
9856-693: 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
9984-417: The area called Oslo (now Gamlebyen or Old Town) was a village or suburb outside the city borders east of Aker river. The population increased from approximately 10.000 in 1814 to 230.000 in 1900. In 1850, Christiania overtook Bergen and became the most populous city in the country. Christiania expanded its industry from 1840, most importantly around Akerselva . There was a spectacular building boom during
10112-428: The capital of the independent Kingdom of Norway, when the union with Denmark was dissolved and replaced by a personal union with Sweden . Several state institutions were established and the city's role as a capital initiated a period of rapidly increasing population. The government of this new state needed buildings for its expanding administration and institutions. Several important buildings and landmarks were erected in
10240-509: The city's economic development, especially before the Black Death. At the end of the 12th century, Hanseatic League traders from Rostock moved into the city and gained major influence in the city. On 25 July 1197, Sverre of Norway and his soldiers attacked Oslo from Hovedøya . During the Middle Ages , Oslo reached new heights during the reign of Haakon V of Norway (1299–1319). He
10368-537: The city's foreign trade in the 15th century. In 1380, Norway was the weaker part in a personal union with Denmark, and Oslo's role was reduced to that of provincial administrative centre, with the monarchs residing in Copenhagen . Over the years, fires destroyed significant parts of the city multiple times, as many of the city's buildings were built entirely of wood. After the fourteenth calamity, in 1624, which lasted for three days, Christian IV of Denmark decided that
10496-428: The city's population is small compared to most European capitals, it occupies an unusually large land area, of which two-thirds are protected areas of forests, hills and lakes. Its boundaries encompass many parks and open areas , giving it an airy and green appearance. Oslo has a humid continental climate ( Köppen climate classification : Dfb ) or, if the original Köppen winter threshold −3 °C (27 °F)
10624-405: The converse holds only if the signature contains no relation symbols, such as in groups or fields. A field or a vector space can be regarded as a (commutative) group by simply ignoring some of its structure. The corresponding notion in model theory is that of a reduct of a structure to a subset of the original signature. The opposite relation is called an expansion - e.g. the (additive) group of
10752-446: The definitions mentioned here are parameter-free , that is, the defining formulas don't mention any fixed domain elements. However, one can also consider definitions with parameters from the model . For instance, in R {\displaystyle \mathbb {R} } , the formula uses the parameter π {\displaystyle \pi } from R {\displaystyle \mathbb {R} } to define
10880-526: The development of model theory throughout its history. For instance, while stability was originally introduced to classify theories by their numbers of models in a given cardinality , stability theory proved crucial to understanding the geometry of definable sets. A first-order formula is built out of atomic formulas such as R ( f ( x , y ) , z ) {\displaystyle R(f(x,y),z)} or y = x + 1 {\displaystyle y=x+1} by means of
11008-440: The equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the semiring of natural numbers N {\displaystyle {\mathcal {N}}} , viewed as a structure with binary functions for addition and multiplication and constants for 0 and 1 of the natural numbers, for example, an element n {\displaystyle n} satisfies
11136-463: The fastest growing major city in Europe at the time. This growth stems for the most part from international immigration and related high birth rates, but also from intra-national migration. By 2010 the immigrant population in the city was growing somewhat faster than the Norwegian population, and in the city proper this had become more than 25% of the total population if the children of immigrant parents were included. The municipality of Oslo has
11264-410: 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
11392-399: The forest. There are two major forests bordering the city: Østmarka (literally "Eastern Forest", on the eastern perimeter of the city), and the very large Nordmarka (literally "Northern Forest", stretching from the northern perimeter of the city deep into the hinterland). The lake's altitude above sea level is 183 metres. The water is in a popular hiking area. Near the water itself, it
11520-509: The formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski , who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah 's stability theory . Compared to other areas of mathematical logic such as proof theory , model theory
11648-679: The formal notion of an " interpretation " of one structure in another). Example: A common signature for ordered rings is σ o r = ( 0 , 1 , + , × , − , < ) {\displaystyle \sigma _{or}=(0,1,+,\times ,-,<)} , where 0 {\displaystyle 0} and 1 {\displaystyle 1} are 0-ary function symbols (also known as constant symbols), + {\displaystyle +} and × {\displaystyle \times } are binary (= 2-ary) function symbols, − {\displaystyle -}
11776-530: The formula φ {\displaystyle \varphi } if and only if n {\displaystyle n} is a prime number. The formula ψ {\displaystyle \psi } similarly defines irreducibility . Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth" , for the satisfaction relation ⊨ {\displaystyle \models } , so that one easily proves: A set T {\displaystyle T} of sentences
11904-490: The formula defines the subset of prime numbers, while the formula defines the subset of even numbers. In a similar way, formulas with n free variables define subsets of M n {\displaystyle {\mathcal {M}}^{n}} . For example, in a field, the formula defines the curve of all ( x , y ) {\displaystyle (x,y)} such that y = x 2 {\displaystyle y=x^{2}} . Both of
12032-403: The function graph is a definable relation, and constants a ∈ M {\displaystyle a\in {\mathcal {M}}} are definable if there is a formula φ ( x ) {\displaystyle \varphi (x)} such that a is the only element of M {\displaystyle {\mathcal {M}}} such that φ (
12160-464: The given language if each sentence in T {\displaystyle T} is true in N {\displaystyle {\mathcal {N}}} with respect to the interpretation of the signature previously specified for N {\displaystyle {\mathcal {N}}} . (Again, not to be confused with the formal notion of an " interpretation " of one structure in another) A model of T {\displaystyle T}
12288-396: The interpreted structures to the language of the original structure. Thus one can show that if a structure M {\displaystyle {\mathcal {M}}} interprets another whose theory is undecidable, then M {\displaystyle {\mathcal {M}}} itself is undecidable. For a sequence of elements a 1 , … ,
12416-493: The invasion, most notably the sinking of the Blücher , delayed the occupation of Oslo for several hours which allowed King Haakon to escape the city. Oslo remained occupied throughout the war until Germany capitulated in 1945. During this time, the occupying troops were harried by saboteurs in acts of resistance. On 31 December 1944, allied bombers missed their intended target and hit a tram, resulting in 79 civilian deaths. During
12544-412: The last decades of the 19th century, with many new apartment buildings and renewal of the city center, but the boom collapsed in 1899. In 1877 the city was renamed Kristiania . The city and municipality used the name Kristiania until 1 January 1925 when the original name of Oslo was restored . This was because Norway became fully independent in 1905, and Norwegians argued that a name memorializing
12672-434: The last overnight freeze (low below 0 °C, 32.0 °F) in spring is 23 April and average date for first freeze in autumn is 17 October giving a frost-free season of 176 days (1981–2010 average for Blindern). Oslo sits right on the border between hardiness zones 7a and 7b. Oslo Gardermoen airport is located 35 km northeast of Oslo and has a humid continental climate (Köppen climate classification: Dfb ) bordering on
12800-425: The model theory of ordered structures. A densely totally ordered structure M {\displaystyle {\mathcal {M}}} in a signature including a symbol for the order relation is called o-minimal if every subset A ⊆ M {\displaystyle A\subseteq {\mathcal {M}}} definable with parameters from M {\displaystyle {\mathcal {M}}}
12928-410: The modern, vastly enlarged Oslo municipality. At the time, Aker was a mostly affluent, green suburban community, and the merger was unpopular in Aker. Other suburbs, such as Lambertseter , began to be developed in the 1950s. Aker Brygge was constructed on the site of the former shipyard , Akers Mekaniske Verksted , from 1982 to 1998. Norway was invaded by Germany on 9 April 1940. Efforts to stop
13056-410: The municipal authorities in 1897, although 'Christiania' was also used. In 1925, the city, after incorporating the village retaining its former name, was renamed 'Oslo'. In 1948, Oslo merged with Aker , a municipality which surrounded the capital and which was 27 times larger, thus creating the modern, much larger Oslo municipality. Oslo is the economic and governmental centre of Norway. The city
13184-536: The name of a large farm at Bjørvika , while the meaning of that name is disputed. Modern linguists generally interpret the original Óslo , Áslo or Ánslo as either "meadow at the foot of a hill" or "meadow consecrated to the Gods ", with both considered equally likely. Erroneously, it was once assumed that Oslo meant "the mouth of the Lo river", a supposed previous name for the river Alna . Not only has no evidence been found of
13312-429: The north and east, wide forested hills ( Marka ) rise above the city giving the location the shape of a giant amphitheatre . The urban municipality ( bykommune ) of Oslo and county ( fylke ) of Oslo are two parts of the same entity, making Oslo the only city in Norway where two administrative levels are integrated. Of Oslo's total area, 130 km (50 sq mi) is built-up and 9.6 km (3.7 sq mi)
13440-567: The northernmost end of the Oslofjord . The fjord , which is nearly bisected by the Nesodden peninsula opposite Oslo, lies to the south; in all other directions Oslo is surrounded by green hills and mountains. There are 40 islands within the city limits, the largest being Malmøya (0.56 km or 0.22 sq mi), and scores more around the Oslofjord. Oslo has 343 lakes, the largest being Maridalsvannet (3.91 km or 1.51 sq mi). This
13568-400: 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
13696-504: The old city should not be rebuilt again. His men built a network of roads on the other side of the bay in Akershagen near Akershus Castle . He demanded that all citizens move their shops and workplaces to the newly built city of "Christiania", named in his honor. The part of the city built starting in 1624 is now often called Kvadraturen [ Wikidata ] because of its orthogonal layout in regular, square blocks. Anatomigården
13824-409: The original structure via an equivalence relation. An important example is a quotient group of a group. One might say that to understand the full structure one must understand these quotients. When the equivalence relation is definable, we can give the previous sentence a precise meaning. We say that these structures are interpretable . A key fact is that one can translate sentences from the language of
13952-510: 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
14080-475: The proof. The completeness theorem allows us to transfer this to satisfiability. However, there are also several direct (semantic) proofs of the compactness theorem. As a corollary (i.e., its contrapositive), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in model theory, where the words "by compactness" are commonplace. Another cornerstone of first-order model theory
14208-409: The reign of King Christian IV , a new city was built closer to Akershus Fortress and named Christiania in honour of the king. It became a municipality ( formannskapsdistrikt ) on 1 January 1838. The city functioned as the capital of Norway during the 1814–1905 union between Sweden and Norway . From 1877, the city's name was spelled Kristiania in government usage, a spelling that was adopted by
14336-414: The river became the symbol of the stable and consistent economic and social divide of the city into an East End and a West End ; the labourers' neighbourhoods lie on both sides of the river, and the divide in reality follows Uelands street a bit further west. River Alna flows through Groruddalen, Oslo's major suburb and industrial area. The highest point is Kirkeberget, at 629 m (2,064 ft). Although
14464-410: The same complete type over A . The real number line R {\displaystyle \mathbb {R} } , viewed as a structure with only the order relation {<}, will serve as a running example in this section. Every element a ∈ R {\displaystyle a\in \mathbb {R} } satisfies the same 1-type over the empty set. This is clear since any two real numbers
14592-409: 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
14720-411: 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
14848-534: The scope of a corresponding quantifier. Examples for formulas are φ {\displaystyle \varphi } (or φ ( x ) {\displaystyle \varphi (x)} to indicate x {\displaystyle x} is the unbound variable in φ {\displaystyle \varphi } ) and ψ {\displaystyle \psi } (or ψ ( x ) {\displaystyle \psi (x)} ), defined as follows: (Note that
14976-408: The set of prime ideals of the polynomial ring A [ x 1 , … , x n ] {\displaystyle A[x_{1},\ldots ,x_{n}]} , and the type-definable sets are exactly the affine varieties. While not every type is realised in every structure, every structure realises its isolated types. If the only types over the empty set that are realised in
15104-612: 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
15232-482: The subset. This generalises the analogous concepts from algebra; for instance, a subgroup is a substructure in the signature with multiplication and inverse. A substructure is said to be elementary if for any first-order formula φ {\displaystyle \varphi } and any elements a 1 , ..., a n of A {\displaystyle {\mathcal {A}}} , In particular, if φ {\displaystyle \varphi }
15360-416: 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
15488-451: The theory of a structure has quantifier elimination, every set definable in a structure is definable by a quantifier-free formula over the same parameters as the original definition. For example, the theory of algebraically closed fields in the signature σ ring = (×,+,−,0,1) has quantifier elimination. This means that in an algebraically closed field, every formula is equivalent to a Boolean combination of equations between polynomials. If
15616-412: The total population of this agglomeration was 1,546,706 in 2023. The city centre is situated at the end of the Oslofjord , from which point the city sprawls out in three distinct "corridors"—inland north-eastwards, and southwards along both sides of the fjord—which gives the urbanized area a shape reminiscent of an upside-down reclining "Y" (on maps, satellite pictures, or from high above the city). To
15744-477: The whole city, while the suburb was renamed "Gamlebyen" , literally "the Old town", to avoid confusion. The Old Town is an area within the administrative district Gamle Oslo . The previous names are reflected in street names like Oslo gate (Oslo street) and Oslo hospital. The origin of the name Oslo has been the subject of much debate. It is nigh-certainly derived from Old Norse and was—in all probability—originally
15872-529: The world for living expenses after Tokyo . In 2013, Oslo tied with the Australian city of Melbourne as the fourth most expensive city in the world, according to the Economist Intelligence Unit (EIU)'s Worldwide Cost of Living study. Oslo was ranked as the 24th most liveable city in the world by Monocle magazine. Oslo's population was increasing at record rates during the early 2000s, making it
16000-726: 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
16128-425: Was even the sunniest year on record with 2133 sunhours. On 27 July 2018 , the temperature in Oslo rose to 34.6 °C (94.3 °F), the hottest recorded since 1937, when weather recordings started at Blindern. In January, on average three out of four days are below freezing (0 °C [32 °F]) and one out of four days is colder than −10 °C (14 °F) (1961–1990). The coldest temperature recorded
16256-574: Was the first king to reside permanently in the city, and it has been regarded as the capital city of Norway since his reign. He also started the construction of the Akershus Fortress and the Oslo Kongsgård . The Black Death came to Norway in 1349 and, like other cities in Europe, the city suffered greatly. The churches' earnings from their land dropped so much that the Hanseatic traders dominated
16384-437: 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). Kristiania Oslo ( Norwegian: [ˈʊ̂ʂlʊ] or [ˈʊ̂slʊ, ˈʊ̀ʂlʊ] ; Southern Sami : Oslove )
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