A theorem established by Leopold Löwenheim (1878–1957) and Thoralf Skolem (1887–1963). The theorem applies to classical first-order logic, and has upward and downward parts. A model of a set of sentences is an interpretation which makes all of its members true. The cardinality of an interpretation is the cardinality of its domain. The cardinality of a language is the cardinality of its set of non-logical symbols. Upward part: if a set of sentences has a model of any infinite cardinality $\kappa$, it has a model of any cardinality greater than $\kappa$. Downward part: if a set of sentences in language $ℒ$ has a model of any infinite cardinality, $\kappa$, it has a model of any cardinality less than $\kappa$ and greater than or equal to the cardinality of $ℒ$. In particular, if, as usual, the language is countable, the set of sentences has a countable model.