Löwenheim number

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.

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.

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 κ_φ be the smallest cardinality of a model of φ, if φ has any model, and let κ_φ 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.

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.

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