|Description: Kuratowski's ordered pair
definition. Definition 9.1 of [Quine] p. 58.
For proper classes it is not meaningful but is well-defined and we allow
it for convenience (see opprc1 3202, opprc1b 3557, opprc2 3203, and opprc3 3558).
For the justifying theorem (for sets) see opth 3545.
There are other ways
to define ordered pairs; the basic requirement is that two ordered pairs
are equal iff their respective members are equal. In 1914 Norbert Wiener
gave the first successful definition _2
, justified by opthwiener 3569, which
was simplified by Kazimierz Kuratowski in 1921 to our present definition.
An even simpler definition _3
justified by opthreg 6033, but it requires the Axiom of Regularity for
justification and is not commonly used. A definition that also works for
proper classes is _4
, justified by
opthprc 4053. If we restrict our sets to nonnegative
integers, an ordered
pair definition that involves only elementary arithmetic is provided by
nn0opthi 8743. Finally, an ordered pair of real numbers
can be represented
by a complex number as shown by crui 7681.|