Re: Kuratowski Ordered Pair

Noel wrote:
Hero wrote:

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.

Let's negate:
(a, b ) = | = ( p, q ) <==? = => a =|= p OR b =|= q
a =|= p OR b =|= q ==> { a, b } =|= { p, q } but it does not follow
that a is before b
.............................

There is possibly one other small consequence. The theory of sets might
have difficulty in maintaining it has a foundation for arithmetic.

( a, a )=|= ( a ), but how can it be that a = a and that a is
existing twofold?
Look at Aristotle for the most basic law of math.

And when ( a, b ) =|= ( b, a ) compare this to the invariant of time.

With friendly greetings
Hero
.