2.6.11: The Löwenheim-Skolem Theorem
The Löwenheim-Skolem Theorem says that if a theory has an infinite model, then it also has a model that is at most countably infinite. An immediate consequence of this fact is that first-order logic cannot express that the size of a structure is uncountable: any sentence or set of sentences satisfied in all uncountable structures is also satisfied in some countable structure.
If \(\Gamma\) is consistent then it has a countable model, i.e., it is satisfiable in a structure whose domain is either finite or countably infinite.
Proof. If \(\Gamma\) is consistent, the structure \(\Struct M\) delivered by the proof of the completeness theorem has a domain \(\Domain{M}\) that is no larger than the set of the terms of the language \(\Lang L\) . So \(\Struct M\) is at most countably infinite. ◻
If \(\Gamma\) is a consistent set of sentences in the language of first-order logic without identity, then it has a countably infinite model, i.e., it is satisfiable in a structure whose domain is infinite and countable.
Proof. If \(\Gamma\) is consistent and contains no sentences in which identity appears, then the structure \(\Struct M\) delivered by the proof of the completness theorem has a domain \(\Domain{M}\) identical to the set of terms of the language \(\Lang L'\) . So \(\Struct{M}\) is countably infinite, since \(\Trm[L']\) is. ◻
Zermelo-Fraenkel set theory \(\Log{ZFC}\) is a very powerful framework in which practically all mathematical statements can be expressed, including facts about the sizes of sets. So for instance, \(\Log{ZFC}\) can prove that the set \(\Real\) of real numbers is uncountable, it can prove Cantor’s Theorem that the power set of any set is larger than the set itself, etc. If \(\Log{ZFC}\) is consistent, its models are all infinite, and moreover, they all contain elements about which the theory says that they are uncountable, such as the element that makes true the theorem of \(\Log{ZFC}\) that the power set of the natural numbers exists. By the Löwenheim-Skolem Theorem, \(\Log{ZFC}\) also has countable models—models that contain “uncountable” sets but which themselves are countable.