Re: Dedekind Cuts, Fundamental Sequences: why?

On Fri, 08 Jun 2007 11:39:02 -0400, Hatto von Aquitanien wrote:
Dave Seaman wrote:

On Fri, 08 Jun 2007 10:38:18 -0400, Hatto von Aquitanien wrote:
Dave Seaman wrote:

On Fri, 08 Jun 2007 03:54:09 -0400, Hatto von Aquitanien wrote:
Dave Seaman wrote:

On Thu, 07 Jun 2007 07:54:50 -0400, Hatto von Aquitanien wrote:

I guess one could argue that even if we do have Cauchy sequences in
which some of the representations have irrational elements, they must
all have representations which are purely rational due to the fact
that the real
numbers were completely defined by the latter. That is, of course,
assuming that such a construction actually took place.

Yes, a sequence of reals has a representation as a sequence of
sequences of rationals.

Is such a sequence of sequences of rational numbers a real number?

No.

What if all the sequences of rational numbers all converge to rational
numbers?

A real number is an equivalence class of Cauchy sequences of rationals. A
sequence of sequences is not an equivalence class of sequences.

So there are no real numbers which are rational numbers?

What did I say that made you think that?

In another recent post I pointed out that an equivalence class of Cauchy
sequences of rationals represents a rational number in our construction,
iff the equivalence class has a constant sequence as one of its members.


--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
.

Relevant Pages

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