Editing Function (mathematics) (section)

== In the foundations of mathematics ==

The definition of a function that is given in this article requires the concept of [[set (mathematics)|set]], since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.

For example, the [[singleton set]] may be considered as a function <math>x\mapsto \{x\}.</math> Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definitions for these weakly specified functions.<ref>{{harvnb |Gödel |1940 |p=16}}; {{harvnb |Jech |2003 |p=11}}; {{harvnb |Cunningham |2016 |p=57}}</ref>

These generalized functions may be critical in the development of a formalization of the [[foundations of mathematics]]. For example, [[Von Neumann–Bernays–Gödel set theory]], is an extension of the set theory in which the collection of all sets is a [[Class (set theory)|class]]. This theory includes the [[Von Neumann–Bernays–Gödel set theory#NBG's axiom of replacement|replacement axiom]], which may be stated as: If {{mvar|X}} is a set and {{mvar|F}} is a function, then {{math|''F''[''X'']}} is a set.

In alternative formulations of the foundations of mathematics using [[type theory]] rather than set theory, functions are taken as [[primitive notion]]s rather than defined from other kinds of object. They are the inhabitants of [[function type]]s, and may be constructed using expressions in the [[lambda calculus]].<ref>
{{cite book
 | last = Klev | first = Ansten
 | editor1-last = Centrone | editor1-first = Stefania
 | editor2-last = Kant | editor2-first = Deborah
 | editor3-last = Sarikaya | editor3-first = Deniz
 | contribution = A comparison of type theory with set theory
 | doi = 10.1007/978-3-030-15655-8_12
 | isbn = 978-3-030-15654-1
 | location = Cham
 | mr = 4352345
 | pages = 271–292
 | publisher = Springer
 | series = Synthese Library
 | title = Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts
 | volume = 407
 | year = 2019}}</ref>