Re: Kuratowski Ordered Pair

On Sun, 02 Dec 2007 04:00:59 +0100, G. Frege <nomail@invalid> wrote:

On Sun, 02 Dec 2007 03:48:02 +0100, G. Frege <nomail@invalid> wrote:


Surely the construction of the Naturals as sets [...] already
presupposes, in an axiomatic approach, the notion of an ordered pair,

No, it doesn't. And IF it would (which is not the case) it would only
presuppose a /Kuratowski-pair/ (or an "elementary pair") --- following
my approach mentioned above.

Note that the successor "function" in set theory actually is introduced
as a set theoretic /operation/:

x' =df x u {x}.

We do not need the notion /ordered pair/ for that.


Just checked the (interesting) Wikipedia entry

http://en.wikipedia.org/wiki/Ordered_pair

There I found the following statement:

"Quine-Rosser definition

Rosser (1953) made extensive use of a definition of the ordered pair due
to Willard van Orman Quine. The Quine-Rosser definition requires a prior
definition of the natural numbers."

See?

Note that this definition doesn't make much (any?) sense in usual set
theories like ZFC:


I suppose it was those 'normal' kinds of set theory I had in mind.
Even there my understanding is probably woefully inadequate. But I confess
I should really have examined the Quine-Rosser approach before making the
original post.

noel
.

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