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In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements are precisely those elements satisfying some formula in the structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining a set. Note that a unary relation on the domain of a structure is simply a subset of the domain, and when we refer to the definable sets in a structure we often mean the definable subsets of the domain. Formally,

Let be the structure consisting of the natural numbers with the usual ordering. Then every natural is definable in without parameters--the number 0 by the formula stating there exist no elements less than me

and every natural n > 0 by the formula stating there exist exactly n elements less than me

In contrast, one cannot define an integer without parameters in the structure consisting of the integers with the usual ordering (this can be proved formally using the automorphism theorem for definable sets; see below).

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