**Description: **Definition of an ordered
pair, equivalent to Kuratowski's definition
when the arguments are
sets. Since the
behavior of Kuratowski definition is not very useful for proper classes,
we define it to be empty in this case (see opprc1 3966, opprc2 3967, and
0nelop 4406). For Kuratowski's actual definition when
the arguments are
sets, see dfop 3943. For the justifying theorem (for sets) see
opth 4395.
See dfopif 3941 for an equivalent formulation using the operation.
Definition 9.1 of [Quine] p. 58 defines
an ordered pair unconditionally
as
, which has different
behavior from our df-op 3783 when the arguments are proper classes.
Ordinarily this difference is not important, since neither definition is
meaningful in that case. Our df-op 3783 was chosen because it often makes
proofs shorter by eliminating unnecessary sethood hypotheses.
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 4418. This was simplified by Kazimierz Kuratowski
in 1921 to
our present definition. An even simpler definition _3
is justified by opthreg 7529, 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 4884. If we restrict our sets to nonnegative
integers, an ordered
pair definition that involves only elementary arithmetic is provided by
nn0opthi 11518. Finally, an ordered pair of real numbers
can be
represented by a complex number as shown by cru 9948.
(Contributed by NM,
28-May-1995.) (Revised by Mario Carneiro,
26-Apr-2015.) |