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Topic: Weierstrass
Replies: 26   Last Post: Jul 9, 2004 4:27 PM

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Jesse F. Hughes

Posts: 9,776
Registered: 12/6/04
Re: Weierstrass
Posted: Jul 9, 2004 1:04 PM
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Eckard Blumschein <blumschein@et.uni-magdeburg.de> writes:

> Jesse F. Hughes wrote:
>

>>>Isn't the mathematical meaning: Zero is included as well as
>>>excluded?

>
> I do not doubt that David W. Cantrell is correct when he tells me that
> the empty set does not include zero if it includes nothing. I just
> wonder how to decide whether it is open or closed if zero is not
> included and not excluded.
>

>> Included and excluded in what? How would that be the meaning?
>
> As a layman, I associate open or closed sets and intervals with excluded
> or included front-numbers, respectively. In N, [4, 7) means 4, 5, 6

^^^^^
Just front or both endpoints? An interval of R is open iff it is of
the form (x,y) for some x and y. Well, even for this, we have to be
careful. We have to interpret (x,y) as the empty set if x > y or
x = y, and we have to include (-oo, oo) = R. For closed sets, it's a
bit more annoying. It's easy to write the emptyset as [x,y] where
x > y, but I don't know how to write R as an interval.

Anyway, an open set may be any union of intervals, so this association
shouldn't be taken overly literally.

> While I am familiar with application of fuzzy logic, I am not aware of
> any awareness, apart from Buridan's donkey, of the fact that infinitely
> precise reals are totally impractical.


Some of the motivation for fuzzy set theory is to develop a set theory
in which one can accommodate fuzzily measured values. Unlike you, the
fuzzy set theorists don't complain that the traditional (classic or
crisp) reals are problematic. Fuzzy set theorists recognize that the
classic theory of real numbers is a perfectly adequate mathematical
theory, but some of them suggest that it's the wrong theory for
physical science. For that, they say, fuzzy reals are preferable.

I do not advocate that view. But it might be of interest to you. I'm
concerned that (1) you may not have the mathematical background to
understand the theory and (2) the fact that some of the theory's
proponents are, uh, exuberant (prone to mentioning Kuhn's theory of
scientific revolution in assessing the importance of their fuzzy sets)
may be a bad influence. Nonetheless, you may wish to check out some
of the theory.

--
Jesse Hughes
"[I]f gravel cannot make itself into an animal in a year, how could it
do it in a million years? The animal would be dead before it got
alive." --The Creation Evolution Encyclopedia




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