Re: Kuratowski Ordered Pair

On Dec 18, 2:34 am, noel etters wrote:
How on earth could you interpret the 'rigmarole' as an argument
about commutivity? You don't seem interested in trying to understand what
I
am saying.

You don't seem intererested in what *I* am saying.

I didn't say that you intend to argue by commutativity. However, I've
pointed out that there is nothing substantive in your argument (or at
least as you presented it) that can't be taken in by noting that
ordered pairing is not commutative.

The point is that the Kuratowski set does not provide a definition
of the ordered pair, as this is understood independently of set theory.

There is no single (let alone rigorous) understanding of ordered pair
independent of a mathematical theory. It's an informal notion and one
that set theory captures in the ways that have been mentioned already.

If your point is that there is some informal notion of which set
theory doesn't capture every possible nuance, shade, connotation, or
other aspect, then, of course, I don't know anyone who claims that a
formal theory can capture all those aspects of an informal notion.
That's a non-starter.

But the Kuratowski defintion does capture the exact DESIRED
mathematical aspects. I've explained that ad nauseam, and will do it
again in this post since you keep skipping past that.

The Kuratowski definition is not INTENDED d to address any host of
other informal notions. But even in informal terms, the Kuratowski
definition is of an operation such that given an input of x and then
y, the operation yields that x is first and that y is second. And the
operation yields that <x y> = <z w> iff x=z and y=w, and that we can
define "first in an ordered pair" and "second in an ordered pair", and
that p is an ordered pair iff p = <1st(p) 2nd(p)>. Again, give me x,
then give me y, and through the Kuratoswki defintion, I give you back
that x is first and that y is second. That there is an "internal"
peculairity in {{x} {x y}} just comes with the territory of
mathematical definition in which one may recognize that, yes, the
PARTICULAR construction may be arbitrary, since there may be more than
one way to capture the"outer" structure we want. It's just finding a
way of "coding" so that our ultimate concern is not in the particular
way we coded but rather in the fact that the coding does capture what
we INTEND to capture.

Instead, it (possibly) provides a definition of a weaker notion, what I
have called a slotted pair. In so far as SS' above (labels for the
distinct
slots) represents an order (since it is a kind of lexical necessity that
we
have either SS' or S'S, or that S and S' taken together be expressed with
some particular orientation), this is the order carried or chosen by the
(arbitrary) decision to go with the Kuratowski main set as opposed to its
reverse

Right there: "as opposed to the reverse". Yes, it is arbitrary, and
that is what I mean by your argument reducing to ordered pairing not
be commutative.

-- it is not defined by the Kuratowski set, it is used by it. In
fact all that is necessary for a slotted pair is that there are understood
to be distinct slots.

None of your rambling about slots and whatnot refutes that the
Kuratowski definition does the job that it is INTENDED to do -
capturing the needed mathematical aspects.

What mathematical result would you like to prove but can't prove due
to some flaw in the Kuratowski definition? What is it mathematically
that you feel you cannot express in the langauge of set theory?

Again,
I confess I may have got carried away in other subthreads in
suggesting tthere might be a huge flaw in set theory because of this. No
doubt the slotted pair is entirely adequate to serve as a building block
for the notions of relation, function and so on. But it is grossly
misleading, at least, to suggest that the Kuratowski set provides a
definition of the ordered pair.

In, say, Z set theories, any set theoretical definition other than the
Kuratowski definition will ARRIVE at the same relevent theorems that I
mentioned above.

We want a definition that yields those relevent theorems.

Since you like to think in metaphor:

Imagine that I give you x and you put some kind of mark on x and then
I give you y and you put a different mark on y. And no matter what two
inputs I give you, you mark them just as you marked x and y. Then you
can always tell me which is first because the first has the mark for
first and the second has the mark for second.

The Kuratowski definition acheives that by (metaphorically speaking
here) putting {} around x as the marker for first and by putting the
first (x) with {} around y to mark that y is second. Granted, it looks
like a rather odd way of doing the marking, but it is efficient and it
acheives exactly what we want of it.

We could use this defintion instead:

<x y> = {{0 x} {1 y}}.

(Would you like that any better?)

Yet, both definitions yield the exact theorems that we INTEND. And
what were interested in is working at the point from which we have
those theorems. How we got our "coding" done to arrive at that point
is not so important.

If you want to axiomatize mathematics in some theory OTHER than set
theory, then, yes, of course, the Kuratowski definition may not be
relevent to your theory at all.

There is no wonder that others, such as
Hero, want to talk about the actual properties of a real ordered pair,

The "actual" properties? A "real" ordered pair? Where may I find this
mysterious creature?

and
no wonder that the Kuratowski definiton generally tends to elicit some
discomfort. But I suspect you have already dismissed any such discomforts
with an airy wave.

I've given detailed accounts. Too detailed. Those are not "airy
waves". Though, I realize that, in your frustration that I don't
happen to agree with you, you would like to condescendignly dismsiss
my actual substantive accounts as "airy waves".

MoeBlee
.

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