Used attributively, originally with reference to the theorem that if a theory in a countable first-order language 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. The second theorem, the downward form of which entails the first theorem, was proved by the Polish mathematician Alfred Tarski 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.
1950s; earliest use found in Journal of Symbolic Logic. From the names of Leopold Löwenheim, German mathematician, and Thoralf Albert Skolem, Norwegian mathematician.