Re: Kuratowski Ordered Pair

On Sun, 02 Dec 2007 02:30:09 +0000, noel etters wrote:


I've always considered that an "ordered pair" should always
be a TWOsome, a doubleton set, never a singleton set;
and also that the two components should have a clear
"unity" and a "duality" about them; and that finally
<a,b> should typically have no b-ness in the 1st
component and no a-ness in the 2nd.

Right. So let's -for the sake of the argument- call

<a, b> := {{a}, {a, b}}

a /Kuratowski-pair/ (or an "elementary pair", etc.).

Then we may use the natural numbers 1 = {0} and 2 = {0,1} to define an
/ordered pair/ the following way:

(a, b) := {<1, a>, <2, b>}.

This set always has two elements, a first (namely <1, ...>) and a second
(namely <2, ...>). Moreover those two components have a clear "one" (for
"first") and a "two" (for "second") resp. "attached" to them. :-)
Finally, no b-ness in the 1st component and no a-ness in the 2nd.

Moreover this "approach" allows for a natural generalization (->
n-tuple).


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.

--------------------------------------------------

But actually, I suggest to proceed the following way:

First define /Kuratowski-pair/ (or /elementary pair/)

<a, b> =df {{a}, {a, b}}.

Define

0 =df {} ,
1 =df {0} ,
2 =df {0,1} .

Note that we can do that WITHOUT defining the (set of) natural numbers,
and/or the notion of /successor/.

Then we can define /ordered pair/:

(a, b) =df {<1, a>, <2, b>}.

Finally we can proceed as usual (but with _my_ ordered pair instead of
the Kuratowski-pair).


Given the Kuratowski definition, then no doubt an alternative
along the lines you suggest can be erected. The question remains: is the
Kuratowski definition any good?

Sure. It's certainly a simple and "direct" approach for defining the
notion "ordered pair" in set theoretic terms (i.e. as a certain set).


F.

--

E-mail: info<at>simple-line<dot>de
.

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