Let $\mathcal{L}$ be a logic. - The *Löwenheim number* of $\mathcal{L}$ is the smallest cardinal $κ$ such that if an arbitrary sentence of $\mathcal{L}$ has any model, the sentence has a model of cardinality no larger than $κ$. **Remark** Löwenheim proved the existence of this cardinal for any logic in which the collection of sentences forms a [set](https://en.wikipedia.org/wiki/Set$(mathematics) "Set (mathematics)") - The *Löwenheim–Skolem number* of $\mathcal{L}$ is the smallest cardinal $κ$ such that if any set of sentences $T$ ⊆ $\mathcal{L}$ has a model then it has a model of size no larger than $\max(|T|, κ)$. - The *Löwenheim–Skolem–Tarski number* of $\mathcal{L}$ is the smallest cardinal such that if $A$ is any structure for $\mathcal{L}$ there is an [elementary substructure](https://en.wikipedia.org/wiki/Elementary$substructure "Elementary substructure") of $A$ of size no more than $κ$. **Remark** This requires that the logic have a suitable notion of "elementary substructure", for example by using the normal definition of a "structure" from predicate logic. - The *Hanf number* of $\mathcal{L}$ is the smallest cardinal $\kappa$ such that whenever a sentence of $L$ has a model of cardinality $\kappa$ then it has arbitrarily large models.