 Löwenheim–Skolem theorem

The Löwenheim--Skolem theorems control the cardinalities of models of a first-order theory without changing its first-order truths.

The Löwenheim–Skolem theorem is a fundamental result in model theory, a branch of mathematical logic. It describes the relationship between first-order logic, models, and cardinalities (sizes) of structures. There are two versions of the theorem: the downward Löwenheim–Skolem theorem and the upward Löwenheim–Skolem theorem.