Re: Kuratowski Ordered Pair

On Mon, 7 Jan 2008 11:09:22 -0800 (PST), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:

On Jan 5, 8:08 pm, noel etters wrote:
On Wed, 2 Jan 2008 18:06:26 -0800 (PST), MoeBlee <jazzm...@xxxxxxxxxxx>
wrote:

If we're not in a theory in the language of set theory then, yes, the
Kuratowski definition may be quite irrelevent. No one disputes that.
But so what?

        You're twisting my words.

I don't intend to. And you don't say in what way you think I have.

It is held that set theory axiomatizes virtually all of ordinary
mathematics. That, for some people, is a part of the reason for
studying set theory. So, in that sense, if we wern't using set theory,
then I agree that we'd have no (or virtually no, or quite little) use
for the Kuratowski definition.

It is no good
explaining that it satisfies the 'characteristic property'. What happens in
contexts in which sets like the Kuratowski are required to be both ordered
pairs and simply that particular kind of set?

WHAT particular kind of set?

        I simply mean a set of the form {{a}. {a, b}}

        The Kuratowski set is an encoding of the ordered pair. (This wasn't
my idea, but I'll use it.)  Yes, it is an effective encoding because it
satisfies the charateristic property (so-called). By 'effective' here we
mean that each distinct ordered pair will have a unique encoding. Another
immediate 'property' of the ordered pair is that (a, b) ~= (b, a) unless
a=b. One could base an encoding on this property, but it would not be
enough to ensure unique encodings. Effective here does not mean that it
captures any properties of the ordered pair. Indeed the meaning of the
ordered pair is irrelevant. It is effective in the same way that encoding
the letters of the English alphabet by the numbers 1 to 26 would be
effective. One can translate an English language statement into numbers
(with appropriate punctuation) and decode it back again. The only problem
that might arise is if we have an English language statement that combines
words and numbers (numerals). We could use number words, but suppose we
want to be able to use number-numerals. My point is that having shown that
the ordered pair can be encoded as a set, it isn't a case of
done-that-forget-it. Every time, in any context, a Kuratowski set appears,
we will have to know or decide there and then whether it functions as an
ordered pair or whether it functions simply as the native set.

I've never needed to make such a decision, as well as I don't know
what "native set" means.

It's hard not to draw the conclusion that you are being perverse.
If I encode 'The Mark of the Beast' as 666, thereafter when I use 666 I may
be referring simply to the native number, eg. the mumber obtained by
multiplying 333 by 2, or I may be referring to the Mark of the Beast.

Sure, every
time an ordered pair appears we can reduce it, replace it by the Kuratowski
set, but everytime a Kuratowski set appears we have to decide which it is,
native set or encoding for ordered pair.

I've never needed to make such a decision.

Everytime a Kuratowski set, i.e. a set having that form, occurs,
you are making that decision.

Have we really got rid ot the
notion of an ordered pair by this means?

Who said anything about "getting rid" of any notions?

Sloppy language on my part. Presumably you think of the Kuratowski
definition as some sort of reduction, a capturing of (the essential
features of) the notion of an ordered pair in terms of sets. But it seems
an odd sort of reduction which is sustained by the choice to interpret a
Kuratowsk (-like) set as an ordered pair, or not, at every turn.

Noel
.

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