Description: Definition of an ordered
pair, equivalent to Kuratowski's definition
{{𝐴}, {𝐴, 𝐵}} 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 4363, opprc2 4364, and
0nelop 4886). For Kuratowski's actual definition when
the arguments are
sets, see dfop 4339. For the justifying theorem (for sets) see
opth 4871.
See dfopif 4337 for an equivalent formulation using the if operation.
Definition 9.1 of [Quine] p. 58 defines
an ordered pair unconditionally
as ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}, which has different
behavior from our df-op 4132 when the arguments are proper classes.
Ordinarily this difference is not important, since neither definition is
meaningful in that case. Our df-op 4132 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
⟨𝐴, 𝐵⟩_2 = {{{𝐴}, ∅}, {{𝐵}}}, justified by
opthwiener 4901. This was simplified by Kazimierz Kuratowski
in 1921 to
our present definition. An even simpler definition ⟨𝐴, 𝐵⟩_3
= {𝐴, {𝐴, 𝐵}} is justified by opthreg 8398, but it requires the
Axiom of Regularity for its justification and is not commonly used. A
definition that also works for proper classes is ⟨𝐴, 𝐵⟩_4
= ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})), justified by
opthprc 5089. If we restrict our sets to nonnegative
integers, an ordered
pair definition that involves only elementary arithmetic is provided by
nn0opthi 12919. An ordered pair of real numbers can also
be represented by
a complex number as shown by cru 10889. Kuratowski's ordered pair
definition is standard for ZFC set theory, but it is very inconvenient
to use in New Foundations theory because it is not type-level; a common
alternate definition in New Foundations is the definition from [Rosser]
p. 281.
Since there are other ways to define ordered pairs, we discourage direct
use of this definition so that most theorems won't depend on this
particular construction; theorems will instead rely on dfopif 4337.
(Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro,
26-Apr-2015.) (Avoid depending on this
detail.) |