Editing Ordered pair (section)

===Wiener's definition===
[[Norbert Wiener]] proposed the first set theoretical definition of the ordered pair in 1914:<ref>Wiener's paper "A Simplification of the logic of relations" is reprinted, together with a valuable commentary on pages 224ff in van Heijenoort, Jean (1967), ''From Frege to Gödel: A Source Book in Mathematical Logic, 1979–1931'', Harvard University Press, Cambridge MA, {{isbn|0-674-32449-8}} (pbk.). van Heijenoort states the simplification this way: "By giving a definition of the ordered pair of two elements in terms of class operations, the note reduced the theory of relations to that of classes".</ref>
<math display="block">\left( a, b \right) := 
\left\{\left\{ \left\{a\right\},\, \emptyset \right\},\,  \left\{\left\{b\right\}\right\}\right\}.</math>
He observed that this definition made it possible to define the [[type theory|types]] of ''[[Principia Mathematica]]'' as sets. ''Principia Mathematica'' had taken types, and hence [[relation (mathematics)|relations]] of all arities, as [[primitive notion|primitive]].

Wiener used <nowiki>{{</nowiki>''b''}} instead of {''b''} to make the definition compatible with [[type theory]] where all elements in a class must be of the same "type". With ''b'' nested within an additional set, its type is equal to <math>\{\{a\}, \emptyset\}</math>'s.