LöwenheimSkolem

The Löwenheim–Skolem theorem, named after Leopold Löwenheim and Thoralf Skolem, is a fundamental result in model theory and first-order logic. It concerns the existence of models of first-order theories in different cardinalities and has two standard forms: downward and upward.

The downward form states that if a theory T in a language L has an infinite model of cardinality κ ≥ |L|, then T has a model of every smaller infinite cardinality λ with |L| ≤ λ ≤ κ. In particular, if the language L is countable and T has any infinite model, then T has a countable model. This form is frequently cited as showing that first-order theories cannot constrain infinite model sizes from above in a strong way.

The upward form states that if T has a model of cardinality κ, then for every cardinal κ' ≥ κ there is a model of T of cardinality κ'. Thus, once a theory has a model of some size, it has models of all larger sizes as well. Together, the theorems imply that first-order theories with infinite models typically have models in a wide range of cardinalities.

Consequences and significance include the ability to show that many theories (such as the theory of arithmetic or real numbers under standard operations) have non-isomorphic models of different sizes, and to establish the existence of countable models for theories in countable languages. The results also give rise to the Skolem paradox, illustrating that a model of set theory can be countable from the outside, while internal to the model it may appear uncountable. The Löwenheim–Skolem theorems are central to model theory and relate closely to compactness and completeness results.

History: Löwenheim proved the downward form in the 1910s, and Skolem extended and clarified the results in the 1920s; the combined statements are now standard in model theory.