SUO: OT: two-element sets vs. ordered pairs (fwd)
A postscript to my last.
Josiah Lee Auspitz
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---------- Forwarded message ----------
Date: Mon, 19 Mar 2001 19:37:01 -0800 (PST)
From: Paul R. Chernoff <chernoff@math.berkeley.edu>
To: lee@sabre.org
Subject: two-element sets vs. ordered pairs
Dear Lee--
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>.
One can then define the ordered triple <a,b,c> to be <a,<b,c>>, and so forth.
--Paul