Re: Kuratowski Ordered Pair

On Mon, 17 Dec 2007 18:09:37 -0800 (PST), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:

On Dec 17, 6:01 pm, noel etters wrote:

Imagine that (a, b) means that a is in one slot and b is in
another
slot. Call it a slotted pair. We could call the slots S and S', but there
is no need for this to be explicit. The slot is created, as it were,
by 'a'
being in it. S and S' would merely be interchangeable labels. If we had
called them S1 and S2, there would be nothing 'firstish' about the slot
S1.
The important thing is that they are distinct slots. So we allow (a, a),
the same object or term or element or whatever being in both slots. If
however a not= b, then clearly (a, b) is going to be different from (b,
a). 'a' being in one slot and 'b' being in the other is different from 'b'
being in that 'first' slot and 'a' in the other. It is also pretty clear
that another slotted pair, (p, q), can only be equal to the first pair by
having the same elements in the same slots, that is by a=p (in slot S) and
b=q (in slot S'). So, everything is ready, I think, for a Kuratowski
definition of the slotted pair.

(a, b) = {{a}, {a, b}}

We can instead have:

(a, b) = {{b}, {a, b}}

And it should be noted that the expression : {{a}, {a, b}},
which equals (a, b) in the standard Kuratowski, is equal to (b, a) in the
reverse Kuratowski. This corresponds to the arbitrary, alternative
labellings: SS' and S'S.

Yes, the ordered pairing operation is not commutative. Just like we
could have defined 'x^y' to be y to the power x rather than the way we
defined it as x to the power y. The operation of exponentiation is not
commuatative and the operation of ordered pairing is also not
commutative.

I think it's clear that the slots are not ordered. This is even
more obvious if you consider n-tuples. All that's required is as many
distinct slots as there are elements (counting repetitions, of course). As
it were, S, S''', S'', S'''' etc. all in a jumble. We could only count (a,
b) as an ordered pair if we agreed in the first place on an order for S,
S'.
But this is not the situation for an ordered pair. In an ordered
pair, (a, b), 'a' is in the first slot; alternatively it is before 'b'.
There is no first and second, nor before and after with a slotted pair, or
Kuratowski set; there are only distinct slots.

You went through all that rigmarole just to say what can be stated as
simply as "the ordered paring operation is not commutative". Yes, it's
not commutative. Lots of operations are not commutative. That's not
reason to think they're not adequately defined.

MoeBlee

How on earth could you interpret the 'rigmarole' as an argument
about commutivity? You don't seem interested in trying to understand what I
am saying.
The point is that the Kuratowski set does not provide a definition
of the ordered pair, as this is understood independently of set theory.
Instead, it (possibly) provides a definition of a weaker notion, what I
have called a slotted pair. In so far as SS' above (labels for the distinct
slots) represents an order (since it is a kind of lexical necessity that we
have either SS' or S'S, or that S and S' taken together be expressed with
some particular orientation), this is the order carried or chosen by the
(arbitrary) decision to go with the Kuratowski main set as opposed to its
reverse -- it is not defined by the Kuratowski set, it is used by it. In
fact all that is necessary for a slotted pair is that there are understood
to be distinct slots.
I confess I may have got carried away in other subthreads in
suggesting tthere might be a huge flaw in set theory because of this. No
doubt the slotted pair is entirely adequate to serve as a building block
for the notions of relation, function and so on. But it is grossly
misleading, at least, to suggest that the Kuratowski set provides a
definition of the ordered pair. There is no wonder that others, such as
Hero, want to talk about the actual properties of a real ordered pair, and
no wonder that the Kuratowski definiton generally tends to elicit some
discomfort. But I suspect you have already dismissed any such discomforts
with an airy wave.

Noel (waving)
.