Re: SUO: OT: two-element sets vs. ordered pairs (fwd)
> From: Paul R. Chernoff <chernoff@math.berkeley.edu>
> ...
> After rereading what I wrote you, I note that although I emphasized
> the distinction between the set {a,b} whose members are a and b, and the
> *ordered pair <a,b>, whose *first* member is a and whose *second* member is b,
> I failed to give Norbert Wiener's clever construction (which appears in
> one of his earliest papers, titled, I believe, "A reduction of the algebra
> of relations to the algebra of sets").
>
> Wiener defines <a,b> to be a certain set, namely, {{a},{a,b}}. In other
> words, <a,b> is a set whose members are themselves sets, viz. the set {a)
> and the set {a,b}. It is then a straightforward exercise to show that
> <a,b> = <c,d> if, and only if, a = c and b = d. This is, of course, precisely
> the key property one would demand of the ordered pair <a,b>.
This definition is usually credited to Kuratowski (and indeed such
constructions are typically called "Kuratowski ordered pairs"), and was
first published in 1921. However, Wiener was indeed the first to realize
that the notion of an ordered pair can be defined in pure set theory, but
his definition (from his 1914 paper "A Simplification of the Logic of
Relations" and his 1913 thesis) is not as elegant, as it involves an
object other than the members of the pair: <a,b> =df {{0,{a}},{{b}}}.
Nonetheless, since he was the first to propose a set theoretic reduction
of the notion, Wiener is often rightly given some share in the credit for
the definition of ordered pair, even though Kuratowski's actual definition
is the one everyone uses.
-chris
--
Christopher Menzel # web: philebus.tamu.edu/~cmenzel
Philosophy, Texas A&M University # net: chris.menzel@tamu.edu
College Station, TX 77843-4237 # vox: (979) 845-8764