Re: Kuratowski Ordered Pair

On Mon, 10 Dec 2007 03:40:34 -0800 (PST), Hero <Hero.van.Jindelt@xxxxxx>
wrote:

Hero wrote:
noel etters wrote:

.......In other words, defining pairs and
tuples in this way bulds in a kind of chiral factor, as if the universe of
sets was either left-hand or right-handed. But this chirality is precisely
the thing to be defined.

It is not chirality which characterizes an ordered pair of math.
....
When the customer is satisfied he will leave with a pair of shoes,
which are two shoes with some kind of ordering, ordered out of a set
of shoes.
....
Another example of this typ of ordering, which is not the ordering
of ordered pairs in math done by a farmer choosing out of his
animals:
{mare, stallion } {ram, ewes }{sow, boar }{rooster(***), hen }


In this thread several definitions of ordered pairs and tuples were
given, the Kuratowski, the Wiener, the G.Frege. They all satisfy the
characteristic property
(a, b ) = ( u, v ) <===> a = u and b = v.
But this property is not sufficent to define ordered pairs, so how can
we conclude that these definitions give us ordered pairs, how do we
know they are equivalent in respect to it?
Here a set-theory- definition of the shoe-pair, not bases on a prior
definition of an ordered pair :
Given the sets A and B with A n B = { } so
{ { a, b } | a e A and b e B } is set of pairs with components/
elements of A u B
with a kind of ordering.
Why are these shoes not ordered pairs in the mathematical sense?

Because it allows (a, b) (eg (left shoe, right shoe) to be equal to
(b, a) (right shoe, left shoe). But obviously you are aware of that. Also
the stipulation that A n B be {} seems unduly restrictive. But actually the
suggestion that the notion of order is located outside the usual
set-theoretic definitions is interesting.

Instead of asking directly what is meant by an ordered pair or
n-tuple, I think it's interesting to ask instead what is involved in the
so-called characteristic property. So we have some definition for an
n-tuple, an ordered triple, say, (a, b, c), which gives us an expression
using a, b, c, which we can write as E[a, b, c], such that:

E[a, b, c] = E[u, v, w]

iff

a=u and b=v and c=w.

What does this mean? Shocking as this may seem, I do not think it
has anything necessarily to do with order. What it guarantees is that we
have a kind of differentiated triple, in the sense that in (a, b, c),
should it turn out that a=b, then there are still two instances of a (or
b). A defintion satisfying the characteristic property has everything to do
with ensuring that (a, b, c) cannot be equal to (a, b), for example, and
nothing directly to do with order.
When we write:

(a, b, c) = (u, v, w)

only if we already understand that a comes first, b second, c third
(likewise for u, v, w) do we know what we are equating with what. The
characteristic property is not enough, or rather it already presupposes the
essential order.
Get away from the use of consecutive letters.

(x, c, p) = (d, r, t)

If the characteristic property were enough, that x=d and c=r and p=t, how
would we know not to write (x, c, p) as, say, (c, x, p), so that

(x., c, p) = (d, r, t) = (c, x, p) = (p, c, x) = (t, d, r) etc

and all the while we understand that x=d and c=r and p=t?

The characteristic property is not a defining characteristic of the
(ordered) tuple, but a trivial and vacuous consequence employing the order.

This might be clearer if we consider some context where order is
already supplied. Consider the C-triple, which consists of consecutive
Naturals, e.g. [3, 4, 5].
Now we might say that two C-triples, [a, b, c] and [u, v, w], are
equal iff a=u and b=v and c=w, but these equalities have nothing to do
with the order, which is rather to do with the relations between a, b and c
with each other (b = a + 1, etc,). As before we could ask how we know not
to write, say, [5, 3, 4] as a C-triple, since it consists of the three
consecutive naturals, 3, 4 and 5? So there is here too the more primitive
order, that a comes before b, b after a and before c, and finally comes c,
and likewise for u, v, w, but the characteristic property cannot even be
stated without presupposing this.

So, I would argue, the n-tuple is really a sequence, even a
sequence in the set-theoretic sense (a definition of the sequence in
numerical terms is not strictly necessary, but it is reasonable to take the
Naturals as the archetypal sequence). In other words, one has the sequence
of positions, 1st, 2nd, 3rd etc., and the values associated with those
positions. Obviously then any attempt to really define an n-tuple purely in
terms of the values is doomed to hopelessness. Perhaps better to say that
the use of these letters a, b, c, as, I suppose, constant or fixed but
indeterminate quantities is inappropriate, and we should describe an
n-tuple as a sequence of frank variables.

I suppose one could ask, if the Kuratowski set (and any other set
like it) fails to define an ordered pair, or indeed an ordered anything,
could it nevertleless serve as the basis for building up the notions of
operator, relation, functions and so on in the axiomatic theory of sets? I
think the answer to that must be No. The Kuratowski set is a curiosity, a
folly. The idea that it has anything to do with order, even in the very
abstract sense required by the theory of sets, is pure illusion. It is
this, I think, which is at the bottom of the too-small suitcase feeling.
There is possibly one other small consequence. The theory of sets might
have difficulty in maintaining it has a foundation for arithmetic.

Noel

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