Re: Kuratowski Ordered Pair

On 26 Nov., 11:25, Noel Etters wrote:
Kuratowski Ordered Pair

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

It is easy to check that:
{ {a}, {a. b} } = { {p}, {p, q} }
iff
a = p and b =q,
which is of course the whole point (the so-called characteristic property
of ordered pairs). an absolute requirement. That this requirement is in
some sense weak is evidenced by the fact there are other set definitions of
the ordered pair.


This is just the same as when you can define real numbers as Dedekind
cuts or via Cauchy sequences.
Or when you can define complex numbers of pairs of real numbers with
a funny multiplication or as R[X] modulo the ideal (X^2+1).
The existence of these constructions allows us to extend our language
by a notion (e.g. of ordered pair) without having to fear that the
notion is void, and this is the whole purpose of it.
Thus we know that we can consider pairs of anything that is
allowed as an element of a set (in other words: we still
don't know how to talk about the ordered pair (class of all sets,
class of all ordinals), no matter which of the various
set theoretic constructions for pair you take).

Even for the set AxB of pairs of elements in A and B, respectively,
you never(?) need the explicit structure of a pair but rather
the universal property. Which already allows one to observe
that (AxB)xC is canonically isomorphic to Ax(BxC) and thus justifies
writing just AxBxC. Just as well AxA (set of pairs) is canonically
isomorphic to A^2 (set of functions from 2 to A).
If it walks a duck ...


and in particular the Kuratowski definition is really a
pair of definitions (an ordered pair?), the Kuratowski itself and its
so-called reverse (an interesting choice of word): { {b}, {a, b} }.

Or: There is a natural isomorphism between AxB and BxA.

The
Kuratowski has a twin, only the twin is like the embarrassing relative who
never gets invited anywhere, and is scarcely even mentioned.



Let me introduce you to the fo-fum pair, which we write as [a, b].
With a fo-fum pair, one element is understood to be fo and the other fum.
(You can take the same element twice, once as fo, once as fum.) An ordinary
pair a-b (which we might write as {a, b}, an unordered pair, but forget
about existing definitions of ordered pairs for the moment), is not fo-fum,
because there is no indication of which is fo and which is fum. Do you
begin to get the idea of a fo-fum pair? I hope not, because it's virtually
meaningless. One way to express it, however, might be as { {a}, {a, b} },
on the understanding that { {b}, {a, b} } would do just as well. Perhaps
the fo-fum pair would make a bit more sense if we could bring the fact
there is this alternative INSIDE the definition somehow. But how to
implement that?
The point being, I suppose, that there is the formal expression or
construct, { {a}, {a, b} }, on the one hand, and interpretations of it on
the other hand. Of course we know what an ordered pair is. As abstractly as
possible, we could say that the ordered pair is this--THEN--that
(throughout it is to be understood that the 'that' can be 'this-again').
Then the claim is that instead of this--THEN--that, this --AND--
this--AND--that--together will do just as well. Will it? Of course,
everything might depend on the use we make of ordered pairs.

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:

(a, (b, c)), or { {a}, {a. {{b}. {b, c}}}} (or some such horrendous
expression, in case I missed a brace)

will be equal to:

(p, (q, r)), or [substitue horrendous expression got by inputting
Kuratowski with p, q, r]

iff (a, b, c) = (p, q, r).

Does that really make (a, (b, c)) a an ordered triple? Why
shouldn't it just be what it is, what it appears to be, an ordered pair
consisting of an element, 'a', first, or on the left, and an ordered pair
on the right? Which indeed happens to have the property that, expressed as
a Kuratowski set, [horrendous set with a, b, c] equals [horrendous set with
p, q, r] iff a = p, b = q and c = r. Call it a composite triple. Another
kind of composite triple would be: ((a, b), c). Here again, it is true that
((a, b), c) or [horrendous set different from the one above got by
inputting Kuratowski] equals ((p, q), r), or [horrendous set with p, q, r}
iff (a, b, c) equals (p, q, r). We already know this because we know left
nesting would have done equally well as right nesting for defining tuples
-- it's just that you can't have both at the same time.
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))


Looks a bit like domino stones ;)

The difference may matter in some *implementations*, e.g.
in computer programs, where one often has to think about whether
to use an array or a singly linked list or a doubly linked list
or a red-black-tree or or or... These sructures are principally
equivalent but may have different effects on the programs efficiency
due to specific access patterns.
Since such considerations are of no concern for
the use of n-tuples in math, we are still in the position that
any (including your) "definition" will do.

In other contexts one might let the natural number 2^a * 3^b
represent the pair (a,b) of natuaral numbers and be happy
to be able to express this within an arithmetical rather
than set-theoretical context. Or let the single number
2^a * 3^b * 5^n represent the n-tuple (a_1, a_2, ..., a_n)
where a_k = a mod (b+k) and 0 <= a_k < b+k.
Such constructions show once again that you can
talk about / represent the concept of pair / sequence within
arithmetical context, but certainly does not provide
any insight into the "inherent nature" of the pair / sequence
concept (on the contrary it rather shows how powerful
arithmetic is).
The same is true for the Kuratowski or any other pair
definition.

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.

This is of course somewhat problematic, since such a sequence is
usually taken to be a map from {1,...,n} to S, hence
a special subset of {1,...,n} x S, hence a set of *pairs*.

Noel

hagman
.

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