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Topic: Obections to Cantor's Theory (Wikipedia article)
Replies: 1,153   Last Post: Jul 31, 2005 5:27 PM

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 The Ghost In The Machine Posts: 1,432 Registered: 12/6/04
Re: 0.99999 is not 1
Posted: Jul 20, 2005 2:00 AM

In sci.logic, Chan-Ho Suh
<suh@math.ucdavis.nospam.edu>
wrote
on Tue, 19 Jul 2005 15:31:44 -0700
<190720051531448592%suh@math.ucdavis.nospam.edu>:
> In article <ateDe.167016\$_o.102331@attbi_s71>, Stephen Montgomery-Smith
> <stephen@math.missouri.edu> wrote:
>

>> I'm in the process of writing an article about
>> objections to the notion that 0.999... is equal to unity,
>> which I plan to contribute to the Wikipedia. I would be interested in
>> having some intelligent feedback. Here's the article so far.
>>
>>

>
> This post has lotsa potential to go wrong....we'll see.

This thread has already gone very wrong :-) -- and has been rehashed
to some extent in other threads, but here's the gist of the problem.

Briefly put, .999... = 1.000... because there are too many problems
otherwise, although it's not entirely inconsistent to hypothesize
that .999... != 1.000..., but one then has to work with expressions
such as the following.

If we still assume that R is a field (and that .999... and 1.000...
are members thereof), then we get this strange little number
d = 1.000... - 0.999... . (If we extend the concept of field
we still have some odd problems with 'd'.)

Briefly put, if d is this difference, then 10d = 10.000... - 9.999...
by multiplying by 10 -- but if we subtract 9 - 9 = 0 from the
right side we get the rather interesting equality

10d = 10.000... - 9.999... = 1.000... - 0.999... = d.

We can also compute d/10 = 0.1000... - 0.0999...; adding 0.9 - 0.9
to the right side, we get a similarly strange equality

d/10 = 0.1000... - 0.0999... = 1.0000... - 0.9999... = d.

There are three ways to attempt a path around this.

[1] d = 0. This gives us standard mathematics.

[2] Define arithmetic differently for infinitesimals, so that
10d = d/10 = d. I'll admit at this point I've no idea
what the details look like, since d - d could be just
about anything. One might end up with something akin
to little-o notation, but I for one can't be sure.

[3] Assume 1 - 10d = 0.999...9990. This quickly gets ridiculous,
especially if one computes things like
(1-d)^2 = 0.999...99800...001 or
(1-d)^2 - (1-d)^3
= 0.999...99800...001 - 0.999...99700...00299...999 = ??!?

I can't call it a contradiction as such but it's certainly
an abuse of the standard use of the ellipsis.

Some will complain that the above isn't rigorous enough; that is
a fair complaint and one can then walk into the realms of
calculus (specifically, limits), and go down a path trod long
before by Cauchy et al.

Others point out this sequence:

x = .999 ...
10x = 9.999...

10x - x = 9x = 9.000...

therefore x = 1.

The main problem with this "proof" is that 10x-x can be either
9.000... or 8.999..., leaving one a bit lost again with
expressions such as 9 - 9d, if x = 1 - d.

And then there are infinite expressions such as d + d + d + ...,
which might be the limit of such things as
lim(n->+oo) sum(i = 1,n) (i/n)^3 (1/n). (This is equal to 1/4
in standard math, of course.)

Or it might be something completely different. At this point I
for one would just want to stick to the standard path,
0.999... = 1.000... .

--
It's still legal to go .sigless.

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7/20/05 Virgil
7/20/05 Stephen Montgomery-Smith
7/25/05 TonyBones
7/25/05 Chris Menzel
7/19/05 Virgil
7/20/05 TonyBones
7/20/05 Virgil
7/25/05 TonyBones
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7/19/05 Virgil
7/19/05 Barb Knox
7/20/05 Virgil
7/19/05 David C. Ullrich
7/19/05 Alec McKenzie
7/19/05 David C. Ullrich
7/19/05 Jesse F. Hughes
7/19/05 Chan-Ho Suh
7/19/05 Ross Finlayson
7/19/05 Jean-Claude Arbaut
7/19/05 Jesse F. Hughes
7/19/05 Alec McKenzie
7/20/05 David C. Ullrich
7/19/05 Timothy Little
7/19/05 Stephen Montgomery-Smith
7/19/05 Alec McKenzie
7/19/05 Stephen Montgomery-Smith
7/19/05 TonyBones
7/19/05 David Kastrup
7/19/05 TonyBones
7/19/05 David Kastrup
7/19/05 TonyBones
7/19/05 David Kastrup
7/20/05 Han de Bruijn
7/20/05 David Kastrup
7/20/05 TonyBones
7/20/05 David Kastrup
7/20/05 Virgil
7/21/05 Peter Webb
7/21/05 Dr. David Kirkby
7/22/05 Ross Finlayson
7/22/05 David Kastrup
7/22/05 malbrain@yahoo.com
7/22/05 David Kastrup
7/22/05 malbrain@yahoo.com
7/23/05 David Kastrup
7/23/05 malbrain@yahoo.com
7/23/05 Virgil
7/23/05 Peter Webb
7/23/05 malbrain@yahoo.com
7/23/05 David Kastrup
7/23/05 malbrain@yahoo.com
7/23/05 Virgil
7/23/05 Han de Bruijn
7/23/05 Virgil
7/25/05 Han de Bruijn
7/25/05 Jack Markan
7/25/05 David Kastrup
7/25/05 Han de Bruijn
7/25/05 David Kastrup
7/25/05 Dr. David Kirkby
7/26/05 Han de Bruijn
7/26/05 Dr. David Kirkby
7/26/05 TonyBones
7/26/05 malbrain@yahoo.com
7/26/05 Virgil
7/27/05 Han de Bruijn
7/25/05 Dr. David Kirkby
7/26/05 Han de Bruijn
7/26/05 TonyBones
7/26/05 Chris Menzel
7/26/05 Virgil
7/27/05 TonyBones
7/27/05 Virgil
7/27/05 Han de Bruijn
7/27/05 Torkel Franzen
7/27/05 Robert Low
7/27/05 Torkel Franzen
7/27/05 Robert Low
7/27/05 Torkel Franzen
7/27/05 Robert Low
7/27/05 Han de Bruijn
7/27/05 Han de Bruijn
7/27/05 Torkel Franzen
7/27/05 Han de Bruijn
7/27/05 TonyBones
7/26/05 TonyBones
7/26/05 malbrain@yahoo.com
7/26/05 Randy Poe
7/26/05 malbrain@yahoo.com
7/27/05 TonyBones
7/27/05 Virgil
7/27/05 TonyBones
7/27/05 Daryl McCullough
7/27/05 Robert Low
7/27/05 Virgil
7/27/05 David Kastrup
7/26/05 Virgil
7/26/05 TonyBones
7/26/05 David Kastrup
7/26/05 malbrain@yahoo.com
7/27/05 TonyBones
7/27/05 Virgil
7/20/05 TonyBones
7/20/05 malbrain@yahoo.com
7/20/05 Virgil
7/20/05 Dik T. Winter
7/20/05 TonyBones
7/20/05 David Kastrup
7/20/05 TonyBones
7/20/05 Daryl McCullough
7/20/05 TonyBones
7/20/05 Daryl McCullough
7/25/05 TonyBones
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7/25/05 Jack Markan
7/25/05 Virgil
7/25/05 Daryl McCullough
7/20/05 Chris Menzel
7/21/05 Virgil
7/20/05 David Kastrup
7/21/05 Barb Knox
7/22/05 Torkel Franzen
7/24/05 Vaughan Pratt
7/24/05 Daryl McCullough
7/30/05 Jesse F. Hughes
7/31/05 Jesse F. Hughes
7/25/05 TonyBones
7/25/05 David Kastrup
7/25/05 Virgil
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7/25/05 malbrain@yahoo.com
7/25/05 stephen@nomail.com
7/25/05 malbrain@yahoo.com
7/25/05 stephen@nomail.com
7/26/05 Dik T. Winter
7/26/05 malbrain@yahoo.com
7/27/05 Dik T. Winter
7/27/05 malbrain@yahoo.com
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7/28/05 stephen@nomail.com
7/28/05 malbrain@yahoo.com
7/28/05 Dik T. Winter
7/28/05 malbrain@yahoo.com
7/28/05 Dik T. Winter
7/28/05 malbrain@yahoo.com
7/29/05 stephen@nomail.com
7/29/05 malbrain@yahoo.com
7/27/05 TonyBones
7/27/05 Virgil
7/25/05 Barb Knox
7/25/05 malbrain@yahoo.com
7/25/05 stephen@nomail.com
7/26/05 malbrain@yahoo.com
7/26/05 stephen@nomail.com
7/26/05 malbrain@yahoo.com
7/26/05 stephen@nomail.com
7/26/05 malbrain@yahoo.com
7/26/05 TonyBones
7/26/05 malbrain@yahoo.com
7/27/05 TonyBones
7/27/05 malbrain@yahoo.com
7/26/05 Virgil
7/26/05 Barb Knox
7/26/05 Dave Rusin
7/27/05 TonyBones
7/27/05 TonyBones
7/27/05 David Kastrup
7/27/05 malbrain@yahoo.com
7/27/05 Barb Knox
7/27/05 Virgil
7/26/05 malbrain@yahoo.com
7/27/05 TonyBones
7/27/05 malbrain@yahoo.com
7/27/05 Virgil
7/20/05 Virgil
7/25/05 TonyBones
7/25/05 Virgil
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7/20/05 Brian Chandler
7/20/05 TonyBones
7/20/05 Daryl McCullough
7/20/05 stephen@nomail.com
7/21/05 Han de Bruijn
7/21/05 Martin Shobe
7/21/05 Han de Bruijn
7/21/05 Martin Shobe
7/21/05 Dik T. Winter
7/21/05 stephen@nomail.com
7/21/05 David Kastrup
7/21/05 Han de Bruijn
7/21/05