**Description: **Kuratowski's ordered pair
definition. Definition 9.1 of [Quine] p. 58.
As the behavior of the usual Kuratowski definition is not very useful
for proper classes, we define it to be empty in this case (see
opprc1 3426, 0nelop 3796, and opprc2 3427). For the justifying theorem (for
sets) see opth 3785. 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 3808,
which was simplified by Kazimierz Kuratowski in 1921 to our present
definition. An even simpler definition
_3
is justified by opthreg 6732, but it requires the
Axiom of Regularity for its justification and is not commonly used. A
definition that also works for proper classes is
_4
, justified by
opthprc 4312. If we restrict our sets to nonnegative
integers, an ordered
pair definition that involves only elementary arithmetic is provided by
nn0opthi 9963. Finally, an ordered pair of real numbers
can be
represented by a complex number as shown by crui 8772. |