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Topic: Finishing up, explaining FLT Proof conclusion
Replies: 45   Last Post: Aug 1, 2001 2:53 AM

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 Peter Percival Posts: 339 Registered: 12/6/04
Re: Finishing up, explaining FLT Proof conclusion
Posted: Jul 25, 2001 7:18 PM
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James Harris wrote:
>
> ...
>
> Since you guys believe in fractions, I've also said at times that
> they're in a "flat" ring, which is just to say that you don't have any
> fractions in it.

What's a ring without fractions in it? Is it a ring with unity in
which, for every x != 1, there is no y such that x*y = 1 or y*x = 1; and
for x = 1 the only such y is 1?

Whatever it is, are you defining "flat ring" to mean "ring without
fractions"?

> The confusion on this issue can be demonstrated by (1+sqrt(3)i)/2,
> which at least one poster BELIEVED was a fraction (hey, it LOOKS like
> a fraction) though it's provably not.

So define "fraction" and prove it. Or define "fraction" and leave
someone else to prove (or otherwise) it.

> Now, some have claimed that you can just use algebraic numbers as a
> grab bag to handle this and all cases like it, but I think my example
> above shows that doesn't work. Besides, I think the definition for
> algebraic numbers is circular and hardly useful anyway.

Why do you think that the definition of algebraic numbers is circular?
"Hardly useful" I have no objection to; if you hardly use them, so be
it. But circular is another matter.

>
> ...
>
> My issue with the current view in the field is that you can talk about
> rings where "factor" is meaningless. If you can do that then what's
> the point of the ring?

Do you mean that in the reals (for instance) everything is a factor of
everything? (Leaving aside zero.) That doesn't make "factor"
meaningless, though it might make it useless.

>
> Since I only need the operations of addition and multiplication, and
> start from a flat ring--integers--there's NO WAY I can end up in a

Is that a definition of "flat ring"? I mean, is "flat ring" just
another name for the ring of integers?

> ring where factors don't matter.
>
> ...
>
> So, let me re-emphasize my point, by asking you a simple question:
>
> Can you start with integers and end up in the field of rationals with
> fractions...your regular old garden variety fraction like 1/2...using
> only addition and multiplication?

The integers are closed under addition and multiplication, so no, you
can't get 1/2 by iterating the ring operations. But then all rings are
closed under the ring operations.

When you write "fraction", do you mean "non-integral rational number"?

>
> If you answer no, then logic will drive you to accept my proof.
>
> If you answer yes, then I want you to demonstrate.
>
> If you refuse to answer "yes" or "no", then you'll say volumes to the
> world which is suddenly very attentive to what's happening here.
>
> Yes--the world--so be honest.
>
> James Harris

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