Editing Mathematical logic (section)

== Model theory ==
{{Main|Model theory}}
'''[[Model theory]]''' studies the models of various formal theories.  Here a [[theory (mathematical logic)|theory]] is a set of formulas in a particular formal logic and [[signature (logic)|signature]], while a [[structure (mathematical logic)|model]] is a structure that gives a concrete interpretation of the theory. Model theory is closely related to [[universal algebra]] and [[algebraic geometry]], although the methods of model theory focus more on logical considerations than those fields.

The set of all models of a particular theory is called an [[elementary class]]; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes.

The method of [[quantifier elimination]] can be used to show that definable sets in particular theories cannot be too complicated. Tarski established quantifier elimination for [[real-closed field]]s, a result which also shows the theory of the field of real numbers is [[decidable set|decidable]].{{sfnp|Tarski|1948}} He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic. A modern subfield developing from this is concerned with [[o-minimal theory|o-minimal structure]]s.

[[Morley's categoricity theorem]], proved by [[Michael D. Morley]],{{sfnp|Morley|1965}} states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e. all models of this cardinality are isomorphic, then it is categorical in all uncountable cardinalities.

A trivial consequence of the [[continuum hypothesis]] is that a complete theory with less than continuum many nonisomorphic countable models can have only countably many. [[Vaught conjecture|Vaught's conjecture]], named after [[Robert Lawson Vaught]], says that this is true even independently of the continuum hypothesis.  Many special cases of this conjecture have been established.