Re: Kuratowski Ordered Pair

On Thu, 29 Nov 2007 11:46:45 -0800 (PST), Hero <Hero.van.Jindelt@xxxxxx>
wrote:

Noel Etters wrote:

Kuratowski Ordered Pair

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

.....
If the Kuratowski were a good definition, might we not expect to
able to generalize it? So that maybe:
(a, b, c) = { {a}, {a, b}, {a, b, c} }.
That doesn't work, nor I think does any extension of any variant
definition work. That is not, of course, the way triples and n-tuples
generally are construed. They are constructed from nestings of ordered
pairs.
So your triple is : (a, (b, c)).
(Or alternatively, ((a, b), c). So long as you stick to one or the other,
which is reminiscent of the fact that you can use the Kuratowski definition
or its twin but not both. We'll stick to right nesting.)
And generally,
(a, b, c, d,....) = (a, (b, (c, (d......))))....
Such nesting of course preserves the characteristic property, so
that without further proof:

...................
So we have two composite triples, (a, (b, c)) and ((a, b), c). We
know, since order is important, that they are not equivalent, that is to
say they will not have the same value (unless a = b =c). But in some broad,
innocent sense they are the same kind of thing i.e. both simple expressions
in a, b, c involving the order bracket with a single nesting. How is it
that in the first case the internal cell walls, as it were, disappear,
magically leaving us with (a, b, c), but that in the second case no such
'chemical reaction' obtains. In fact we have SACRIFICED the ability to
represent an ordered pair which consists, in the right place, of an ordered
pair. In such constructions the internal brackets around the ordered pair
dissolve, as if on some genetic instruction.

There is a better way to define the n-tuple. I haven't seen this
anywhere, but it seems obvious enought o have been thought of my somebody.

(a, b, c, d......n) =(defn) ((a, b), (b, c), (c, d)...........(m, n))

The n-tuple is an ordered chain of ordered pairs, in which the R
element of a pair is identical to the L element of the next pair.

This is recursive since an n-tuple is an (n - 1)-tuple of ordered
pairs.

In other words,

n-tuple = (X, Y), where X and Y are ordered pairs of ordered pairs of
.....

The quadruple as an example:

(a, b, c, d) = ((a, b), (b, c), (c, d))

= (P, Q, R)

where P = (a, b), Q = (b, c), R = (c, d)

= ((P, Q), (Q, R))

= (X, Y)

where X = (P, Q) = ((a, b), (b, c))
Y = (Q, R) = ((b, c), (c, d))

Note that if we have a definition of the ordered pair, such as the
Kuratowski, the characteristic property for n-tuples will be automatically
preserved here as for the usual nesting definition.

Why is this definition better? Because we no longer have anything
like that peculiar assymetry we had before, that ((a, b), c) and (a, (b,
c)) were not only inequivalent but radically dissimilar. We still have a
kind of dissolution of brackets but only where there is a contiguous
repetition of an element across pairs, in which case there is a
substitution of the repetition by a single instance of the element. It is
clearly quite intuitive. (It's interesting to look at how far one can get
with UNordered pairs which CAN be arranged as a chain as above. But a
definition based on this is not quite sufficient, and in any case we run
into the problem of mult-sets.)
There is still a sacrifice. Baldly stated, we have lost the ability
to represent (without reduction to something else) an ordered pair of
ordered pairs with contiguous repetition across the pairs. More tellingly
put, how is it that ((a, b), (c, d)) should become something of a quite
different form -- i.e. (a, b, c) -- because it happens that b equals c?
Think of a context, if it helps, not that there has to be any particular
context.
It is impossible to get away from the artifice of all this. As if
to say, we CAN reduce n-tuples to pairs (and then on down to sets), but why
SHOULD we? And as I have tried to suggest, this reduction is not without
cost, or internal strain.

What might be at stake here? In the theory of sets the defintion of
sequence is explicitly numerical. Not all sequences, in the ordinary sense
of the term, are numerical. Is the alphabet numerical? Is the bidding
sequence in Bridge (that one club is overbid by one diamond which is
overbid by one heart etc.)? Is Eeny Meeny Miny Mo numerical? Number is
special; it's a very versatile instrument. But that does not make it
fundamental to all notions of order, of sequence. Giving substance to the
notion of an ordered pair holds out the hope of doing justice to such
ideas. But I doubt anyone really believes that is what is going on, and
it's probably stupid to think the theory of sets seeks to make contact with
reality in so crude a manner. Is there anyone who on first aquaintance with
the Kuratowski set does not feel a sense of things being forced, as if one
is trying to cram everything into a suitcase which is really too small for
the job. To my mind what's at work is a legoland fantasy that everything is
made out of simple bullding blocks. It's metaphysical reductionism, and
it's stale as old toast. Appearance and Reality. Everything mathematical is
really logic and sets. Yeah, but your shirts are still going to be creased
when you get them out. And maybe the world as it appears, full of human
quarrels and wars, is really a cosmic battlefield between divine and
satanic forces. Do you really want to pin the theory of sets on this
misshape, the Kuratowski set, and this contrivance of nesting (or chaining)
pairs?

An obvious and at least sensible way to define an ordered pair
(n-tuple) is as a sequence with two (n) members. Whiich, I should imagine,
limits one's options somewhat, axiomatically speaking.


So it is impossible, to define an ordered pair with set-theory only.

Somehow I doubt that it can be done in any satisfactory or perspicuous way.
I never got my head properly around the Quine approach, so the other posts
in this sub-thread were of interest.

Thanks,

Noel
.

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