In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds. They are named after Leopold Löwenheim, who proved that these exist for a very broad class of logics.

Abstract logic

An abstract logic, for the purpose of Löwenheim numbers, consists of: The theorem does not require any particular properties of the sentences or models, or of the satisfaction relation, and they may not be the same as in ordinary first-order logic. It thus applies to a very broad collection of logics, including first-order logic, higher-order logics, and infinitary logics.

Definition

The Löwenheim number of a logic L is the smallest cardinal κ such that if an arbitrary sentence of L has any model, the sentence has a model of cardinality no larger than κ. Löwenheim proved the existence of this cardinal for any logic in which the collection of sentences forms a set, using the following argument. Given such a logic, for each sentence φ, let κφ be the smallest cardinality of a model of φ, if φ has any model, and let κφ be 0 otherwise. Then the set of cardinals exists by the axiom of replacement. The supremum of this set, by construction, is the Löwenheim number of L. This argument is non-constructive: it proves the existence of the Löwenheim number, but does not provide an immediate way to calculate it.

Extensions

Two extensions of the definition have been considered: For any logic for which the numbers exist, the Löwenheim–Skolem–Tarski number will be no less than the Löwenheim–Skolem number, which in turn will be no less than the Löwenheim number. Note that versions of these definitions replacing "has a model of size no larger than" with "has a model smaller than" are sometimes used, as this yields a more fine-grained classification.

Examples