In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. Preliminaries For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language Lσ that can be used to capture the first-order expressible facts about the σ-structure. There are two common ways to specify theories: List or describe a set of sentences in the language Lσ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in Lσ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. An Lσ theory may: be consistent: no proof of contradiction exists; be satisfiable: there exists a σ-structure for which the sentences of the theory are all true (by the completeness theorem, satisfiability is equivalent to consistency); be complete: for any statement, either it or its negation is provable; have quantifier elimination; eliminate imaginaries; be finitely axiomatizable; be decidable: There is an algorithm to decide which statements are provable; be recursively axiomatizable; be model complete or sub-model complete; be κ-categorical: All models of cardinality κ are isomorphic; be stable or unstable; be ω-stable (same as totally transcendental for countable theories); be superstable have an atomic model; have a prime model; have a saturated model. Pure identity theories The signature of the pure identity theory is empty, with no functions, constants, or relations. Pure identity theory has no (non-logical) axioms. It is decidable. One of the few interesting properties that can be stated in the language of pure identity theory is that of being infinite. This is given by an infinite set of axioms stating there are at least 2 elements, there are at least 3 elements, and so on: ∃x1 ∃x2 ¬x1 = x2, ∃x1 ∃x2 ∃x3 ¬x1 = x2 ∧ ¬x1 = x3 ∧ ¬x2 = x3,... These axioms define the theory of an infinite set. The opposite property of being finite cannot be stated in first-order logic for any theory that has arbitrarily large finite models: in fact any