Editing First-order logic (section)

===First-order theories, models, and elementary classes===
A ''first-order theory'' of a particular signature is a set of [[axiom]]s, which are sentences consisting of symbols from that signature. The set of axioms is often finite or [[recursively enumerable]], in which case the theory is called ''effective''. Some authors require theories to also include all logical consequences of the axioms. The axioms are considered to hold within the theory and from them other sentences that hold within the theory can be derived.

A first-order structure that satisfies all sentences in a given theory is said to be a ''model'' of the theory. An ''[[elementary class]]'' is the set of all structures satisfying a particular theory. These classes are a main subject of study in [[model theory]].

Many theories have an ''[[intended interpretation]]'', a certain model that is kept in mind when studying the theory. For example, the intended interpretation of [[Peano arithmetic]] consists of the usual [[natural number]]s with their usual operations. However, the Löwenheim–Skolem theorem shows that most first-order theories will also have other, [[nonstandard model]]s.

A theory is ''[[consistency|consistent]]'' (within a [[First-order logic#Deductive_systems|deductive system]]) if it is not possible to prove a contradiction from the axioms of the theory. A theory is ''[[complete theory|complete]]'' if, for every formula in its signature, either that formula or its negation is a logical consequence of the axioms of the theory. [[Gödel's incompleteness theorem]] shows that effective first-order theories that include a sufficient portion of the theory of the natural numbers can never be both consistent and complete.