Löwenheim-Skolemn.
/ˌləːvənhʌɪmˈskuːləm/, /ˌləʊənhʌɪmˈskəʊləm/, U.S. /ˌləvənhaɪmˈskuləm/, /ˌloʊənhaɪmˈskoʊləm/ Mathematics and Logic.
Used attributively, originally with reference to the theorem that if a theory in a countable first-order language (cf. first-order adj.) has any models, then it has a model with countably many elements; now more commonly with reference to the theorem (also known as the upward and downward Löwenheim-Skolem theorem) that (i) every infinite structure has elementary extensions with all possible cardinalities (known as the upward part of the theorem), and (ii) for every structure M and set X of elements of M, and every possible cardinality, there is an elementary substructure of M which contains all of X and has that cardinality (known as the downward part of the theorem).The first theorem was proved by Skolem 1920, extending earlier ideas of Löwenheim (see reference in etymology). The second theorem, the downward form of which entails the first theorem, was proved by the Polish mathematician Alfred Tarski (see Compositio Mathematica 13 (1957) 92f) and is also occasionally known as the Löwenheim-Skolem-Tarski theorem or the upward and downward Löwenheim-Skolem theorem. The name Löwenheim-Skolem theorem is also extended to various similar theorems, for example theorems stating analogous facts about non-first-order languages.
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the mind > mental capacity > philosophy > logic > predicate or propositional logic > [noun] > mathematical or symbolic logic > theorems, etc.
Löwenheim-Skolem1950
1948 Jrnl. Symbolic Logic 13 47 By use of the Skolem-Löwenheim theorem.]
1950 Jrnl. Symbolic Logic 15 114 Standard theorems such as the Löwenheim-Skolem theorem, that if a logic is consistent it must have a denumerable model, are readily carried out in any of the familiar systems of logic.
1967 J. van Heijenoort From Frege to Gödel 582 His proof yields, besides completeness, the Löwenheim-Skolem theorem, which states that a satisfiable formula is ℵ0-satisfiable.
1974 L. Henkin & J. D. Monk in Proc. Tarski Symp. 112 These steps are accomplished by algebraic versions of the upward and downward Löwenheim-Skolem theorems.
1996 M. Manzano Extensions First-Order Logic vi. 225 It is also proved that first order logic is again the strongest logic with a finitary syntax to possess the Lowenheim-Skolem property and be complete.
2011 Bull. Symbolic Logic 17 382 By the Löwenheim-Skolem Theorem we get Skolem's paradox.
This entry has been updated (OED Third Edition, September 2013; most recently modified version published online March 2022).
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n.1950