Editing First-order logic (section)

===The Löwenheim–Skolem theorem===
The [[Löwenheim–Skolem theorem]] shows that if a first-order theory of [[cardinality]] λ has an infinite model, then it has models of every infinite cardinality greater than or equal to λ. One of the earliest results in [[model theory]], it implies that it is not possible to characterize [[countable set|countability]] or uncountability in a first-order language with a countable signature. That is, there is no first-order formula φ(''x'') such that an arbitrary structure M satisfies φ if and only if the domain of discourse of M is countable (or, in the second case, uncountable).

The Löwenheim–Skolem theorem implies that infinite structures cannot be [[categorical theory|categorically]] axiomatized in first-order logic. For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum. Since the real line is infinite, any theory satisfied by the real line is also satisfied by some [[nonstandard model]]s. When the Löwenheim–Skolem theorem is applied to first-order set theories, the nonintuitive consequences are known as [[Skolem's paradox]].