Re: Kuratowski Ordered Pair

On Sat, 15 Dec 2007 20:16:26 -0800 (PST), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:

On Dec 15, 6:18 pm, noel etters wrote:
On Fri, 14 Dec 2007 14:57:01 -0800 (PST), MoeBlee <jazzm...@xxxxxxxxxxx>
wrote:

On Dec 13, 4:53 am, Hero <Hero.van.Jind...@xxxxxx> wrote:

but it does not follow
that a is before b

We can define "1st of <a b>" and "2nd of <a b>" so that 1st of <a b>
is a and 2nd of <a b> is b, and then "a is before b in pair" as "a is
the 1st of <a b> and b is the 2nd of <a b>".

That's OK.

The Kuratowski definition satisfies the motivation for an ordered pair
operation and mathematics seems to do just fine with the Kuratowski
definition to go on to define relations, orderings, Cartesian
products, sequences, et. al.

How does that work?

If <a, b> is an ordered pair, then:
<a, b> = <p, q> iff a=p and b=q.

You don't need "If <a b> is an ordered pair" there, since we have
simply:

<a b> = <p q> <-> (a=p & b=q)

This is obviously true for an ordered pair, and still true if we write it
as the Kuratowski set. But there is no converse. That is to say, we cannot
derive the notion of an ordered pair from the characteristic property.

If I understand what you're getting at, then, yes, there are ways
other than the Kuratowski defintion to capture the characteristic
property. So? That's the way it is in mathematics with all kinds of
things: natural numbers, cardinality, et. al. We have a notion we wish
to express mathematically, and different formulations will do the job,
so we pick one.

From some undefined notion of pair, but given that:

(a, b) = (p, q) iff a=p and b=q,

it does not follow that (a, b) and (p, q) are ordered pairs, that (a, b) =
<a, b>.

Then you're just arguing from some undefined notion of (a b). But,
again, if I understand your point, yes, it's true that not only the
Kuratowski defintion will work. I don't see what problem there is.

From an as yet undefined pair (x, y), how do we know not to write:
(a, b) = (b, a) = (p, q) = (q, p), while simultaneously and consistently
holding that a=p and b=q?

The Kuratowski definition is really this:

<1st term, 2nd term> = {{1st term}, {1st term, 2nd term}}

Wrong, since "1st" and "2nd" are defined FROM having first made the
definition of ordered pair (whether the Kuratowski definition or some
other).

The (Kuratowski) definition is:

<x y> = {{x} {xy}}.

Then

<x y> = <p q> <-> (x=p & y=q)

is a THEOREM that comes from the definition

Then '1st(p)' and '2nd(p)' are defined, then another theorem:

p is an ordered pair <-> p = <1st(p) 2nd(p)>

so

p is an ordered pair <-> p = {{1st(p)} {1st(p) 2nd(p)}}.

Without this, indeed, we could not even say the Kuratowski
definition gives a different result for <x. y> and <y, x>, for (x, y) = (y,
x) = {{x}, {x, y}} could be consistent as far as we know.

We DO know that it is NOT the case that {{x} {x y}} = {{y} {x y}}
unless x=y. I don't see any point you're making.


So what kind of definition of an ordered pair is this?

A correctly formed definition of a 2-place operation symbol that works
to give us exactly what we want, which is the aforementioned theorems.

We're not
really defining the 'order' bit, since we're just carrying over the 1st,
2nd notion.

Then you'd have to say what "really is the order bit".

Come to think ot it, the 'pair' aspect seems pretty well
covered too by the same terms. So what part of 'ordered pair' does this
definition understand?

I don't even know what you mean by a "defintion" that "understands".

Of course the order in which objects or terms appear is extrinsic
to the objects or term themselves. The ordered pair <apple, orange> could
just as easily have been <orange, apple>. In re-writing the mathematical
object <apple, orange> are we not giving the impression that it becomes a
(mathematical object) complex set involving apples and oranges, when as a
matter of fact the set in question is a set involving 'first-apple's and
'second-orange's?

I have to say that I find it difficult to understand such worry when
the definition works mathematically just as we wish it to work.

What is the thinking here? That we can re-write the ordered pair as
a complex set

What is a "complex set"?

(even though we are not actually defining either the order or
the pair aspect of an ordered pair}. Sets are these very abstract and
versatile mathematical objects we have lots of axioms and theorems for.
That must be worth doing then.
Except that conceptually this is about as convincing as the
spiritual credentials of a stripper dressed up as a nun.

I don't know what it is you want us to be "convinced" of by a
definition, in particular this definition. It works to do the
mathematics we'd like to do. I don't see that it needs to "convince"
us of anything else. We do recognize that it is only one of many ways
we could work it; that the Kuratowski definition formalizes a certain
general notion but is not the SAME as that notion. Well, that's
formalization for you. If one demanded that every formalization were
the same as what it formalizes, then we wouldn't need formalization
anyway.

Try and meet me half way, at least.
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.

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.

Noel


.

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