Editing Ordered pair (section)

===Morse definition===
[[Morse–Kelley set theory]] makes free use of [[proper class]]es.<ref>{{cite book |last=Morse |first=Anthony P. |year=1965 |title=A Theory of Sets |url=https://archive.org/details/theoryofsets0000mors |url-access=registration |publisher=Academic Press }}</ref> [[Anthony Morse|Morse]] defined the ordered pair so that its projections could be proper classes as well as sets. (The Kuratowski definition does not allow this.) He first defined ordered pairs whose projections are sets in Kuratowski's manner. He then ''redefined'' the pair
<math display="block"> (x, y) = (\{0\} \times s(x)) \cup (\{1\} \times s(y))</math>
where the component Cartesian products are Kuratowski pairs of sets and where
<math display="block"> s(x) = \{\emptyset \} \cup \{\{t\} \mid t \in x\} </math>

This renders possible pairs whose projections are proper classes. The Quine–Rosser definition above  also admits [[proper class]]es as projections. Similarly the triple is defined as a 3-tuple as follows:
<math display="block"> (x, y, z) = (\{0\} \times s(x)) \cup (\{1\} \times s(y)) \cup (\{2\} \times s(z))</math>

The use of the singleton set <math> s(x) </math> which has an inserted empty set allows tuples to have the uniqueness property that if ''a'' is an ''n''-tuple and b is an ''m''-tuple and ''a'' = ''b'' then ''n'' = ''m''.  Ordered triples which are defined as ordered pairs do not have this property with respect to ordered pairs.