Definable set

In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation. Let be a first-order language, an -structure with domain , a fixed subset of , and a natural number. Then: A set is definable in with parameters from if and only if there exists a formula and elements such that for all , if and only if The bracket notation here indicates the semantic evaluation of the free variables in the formula. A set is definable in without parameters if it is definable in with parameters from the empty set (that is, with no parameters in the defining formula). A function is definable in (with parameters) if its graph is definable (with those parameters) in . An element is definable in (with parameters) if the singleton set is definable in (with those parameters). Let be the structure consisting of the natural numbers with the usual ordering. Then every natural number is definable in without parameters. The number is defined by the formula stating that there exist no elements less than x: and a natural number is defined by the formula stating that there exist exactly elements less than x: In contrast, one cannot define any specific integer without parameters in the structure consisting of the integers with the usual ordering (see the section on automorphisms below). Let be the first-order structure consisting of the natural numbers and their usual arithmetic operations and order relation. The sets definable in this structure are known as the arithmetical sets, and are classified in the arithmetical hierarchy. If the structure is considered in second-order logic instead of first-order logic, the definable sets of natural numbers in the resulting structure are classified in the analytical hierarchy. These hierarchies reveal many relationships between definability in this structure and computability theory, and are also of interest in descriptive set theory.