Re: SUO: OT: two-element sets vs. ordered pairs (fwd)
>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.
This is just one brick in the wall of ingenious reconstructions of
mathematics in set theory which was built on the foundation provided
by Russell & Whitehead. The upshot of all this ingenuity is that
almost anything that can be said in mathematics can, if challenged as
to its underlying consistency, refer to the challenge to the question
of the consistency of set theory. ("Almost" because some aspects of
category theory don't fit onto set theory very easily, and category
theory has itself been proposed as a more appropriate foundation for
mathematical consistency. There has been a kind of religious struggle
for the soul of mathematics, with Saunders MacLane as its Martin
Luther.)
Apart from providing this consistency proof, however, the
set-theoretic reconstruction isn't all that interesting from a
mathematical point of view. In particular, it would be a mistake to
assume that ordered pairs *really are* sets of the form {{a}{a b}},
or that integers *really are* sets of the form {},{{}},{{{}}},... or
of the form {},{{}},{{}{{}}},{{}{{}}{{}{{}}}},... or whatever. They
might be those things; but they might also be something else
altogether, and the mathematics works just fine without committing
itself to the ultimate nature of the things it is talking about.
Pat Hayes
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