Re: The horse race
Date: 22 Feb 2006 13:13:28 -0800
Message-ID: <1140642808.784901.64510_at_g44g2000cwa.googlegroups.com>
Mark Johnson wrote:
>
> Well, I can understand how one would say that some ordinal attribute
> really is not ordering anything, but simply acts as some sort of
> weighting attribute. But I think at some point that becomes difficult
> to say. At what threshold does the weight have to be termed, a sort?
How many times do you want me to say "there is no such threshold." The answer to your question is "never." By definition.
> But the use terms ought to be. That's the subject, not "Math".
When you're asking about the essential characteristics of relations, the subject is indeed "math."
> Not even a hierarchy, necessarily. Just a sorted list, nothing more.
> Fail to account, throw out that information, and what you have may
> become useless, mathematically or otherwise.
History has shown that sets are useful, so your hypothesis is
incorrect.
Anyway, set theory doesn't discount order; it has lots of useful
things to say about order. I've mentioned a number of them.
> Perhaps for other purposes that set may be unordered.
That's right.
> If I give you page one, with story A in column left and story B
> beginning over in column right, with A paragraph one following the
> headlines and all falling below the banner, and A paragraph three
> following two and preceding four, one might say the set of all
> paragraphs is an unordered set. But there is an intrinsic order.
Yes. But that order is not part of the set of paragraphs S. We may capture that order in an order relation R. The pair (S, R) is called an ordered set. The set S is not called ordered; it is unordered. R is also unordered. R and S are sets; they are unordered by definition. But R specifies a (not "the") order for set S.
Marshall Received on Wed Feb 22 2006 - 22:13:28 CET