Used attributive, 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).