## First-Order Logic and Infinite Models

First-order logic supports the notion of infinite models. In fact, the Löwenheim–Skolem theorem is a fundamental theorem in mathematical logic that guarantees the existence of infinite models for any first-order theory that has an infinite model.

The Löwenheim–Skolem theorem states that if a first-order theory has an infinite model, then it has a model of every infinite cardinality. This means that if a first-order theory has any infinite model, it also has models that are countably infinite, uncountably infinite, or of any other infinite cardinality.

The theorem has important implications for the study of first-order logic, as it shows that the existence of infinite models cannot be used to distinguish between different first-order theories. This is because any first-order theory that has an infinite model has models of every infinite cardinality, which means that the notion of "size" or "complexity" of a model cannot be used to compare different first-order theories.