Editing Setoid (section)

==Type theory==
In type-theoretic foundations of mathematics, setoids may be used in a type theory that lacks [[quotient type]]s to model general mathematical sets. For example, in [[Per Martin-Löf]]'s [[intuitionistic type theory]], there is no type of [[real number]]s, only a type of [[regular Cauchy sequence]]s of [[rational number]]s. To do [[real analysis]] in Martin-Löf's framework, therefore, one must work with a ''setoid'' of real numbers, the type of regular Cauchy sequences equipped with the usual notion of equivalence.  Predicates and functions of real numbers need to be defined for regular Cauchy sequences and proven to be compatible with the equivalence relation.  Typically (although it depends on the type theory used), the [[axiom of choice]] will hold for functions between types (intensional functions), but not for functions between setoids (extensional functions).{{clarify|date=October 2010}} The term "set" is variously used either as a synonym of "type" or as a synonym of "setoid".<ref>{{cite web|page=9|url=http://www.cs.chalmers.se/Cs/Research/Logic/TypesSS05/Extra/palmgren.pdf|title=Bishop's set theory}}</ref>