Löwenheim–Skolem theorem

In mathematical logic, the Löwenheim-Skolem theorem states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have exactly one model up to isomorphism.

Reference:
http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem