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 4660, opprc2 4661, and
0nelop 5191). For Kuratowski's actual definition when
the arguments are
sets, see dfop 4635. For the justifying theorem (for sets) see
opth 5176.
See dfopif 4633 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 4405 when the arguments are proper classes.
Ordinarily this difference is not important, since neither definition is
meaningful in that case. Our df-op 4405 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 5211. 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 8810, 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 5413. Nearly at the same time as Norbert Wiener,
Felix Hausdorff
proposed the following definition in "Grundzüge der
Mengenlehre"
("Basics of Set Theory"), p. 32, in 1914: ⟨𝐴, 𝐵⟩_5
= {{𝐴, 𝑂}, {𝐵, 𝑇}}. Hausdorff used 1 and 2
instead of 𝑂 and 𝑇, but actually any two
different fixed sets
will do (e.g., 𝑂 = ∅ and 𝑇 = {∅}, see 0nep0 5070).
Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different
from 𝐴 as well as 𝐵, which is actually not
necessary (at least
not in full extent), see opthhausdorff0 5215 and opthhausdorff 5214. If we
restrict our sets to nonnegative integers, an ordered pair definition
that involves only elementary arithmetic is provided by nn0opthi 13375. An
ordered pair of real numbers can also be represented by a complex number
as shown by cru 11366. 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 4633.
(Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro,
26-Apr-2015.) (Avoid depending on this
detail.) |