LöwenheimSkolemtételek

LöwenheimSkolemtételek, known in English as the Löwenheim–Skolem theorems, are two fundamental results in model theory about the sizes of models of first-order theories. They are named after the logicians Leopold Löwenheim and Thoralf Skolem, who contributed to their development in the early 20th century. The theorems reveal that first-order theories cannot control the cardinalities of all their models and, in particular, that theories with infinite models must have models of various infinite sizes.

The downward Löwenheim–Skolem theorem states that any theory T in a countable first-order language that has an infinite model also has a countable model. More precisely, if M is a model of T with infinite cardinality, there exists a substructure N of M, with N ≺ M, such that N is countable. This implies that even theories intended to describe large structures can have small, countable realizations.

The upward Löwenheim–Skolem theorem says that if a theory T has a model of some infinite cardinality κ, then it has models of every larger infinite cardinality λ ≥ κ. Together with the downward version, the theorems show that many theories are not categorical in any infinite cardinality; that is, they do not have a unique model up to isomorphism across all infinite sizes.

Historically, the theorems helped illuminate Skolem’s paradox and the limits of first-order logic for uniquely characterizing mathematical structures. They remain central in model theory, influencing topics such as elementary substructures, completeness, and the study of model sizes in logic.