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Topic: 1.28 - The myth of the 'real' number line.
Replies: 38   Last Post: Jul 11, 2014 1:53 PM

 Messages: [ Previous | Next ]
 Michael Press Posts: 2,137 Registered: 12/26/06
Re: 1.28 - The myth of the 'real' number line.
Posted: Jun 23, 2014 7:27 PM

In article <lni2vc\$jvc\$1@news.albasani.net>,
Peter Percival <peterxpercival@hotmail.com> wrote:

> John Gabriel wrote:
> >
> > So, at some time during [snip]

>
> I didn't pay close attention to what I've snipped but I formed the
> impression that you think rational numbers exist, and irrational numbers
> don't. What, then, do you think of the definitions of real numbers in
> terms of rational numbers? (Cuts, Cauchy sequences, nested intervals.)
>
> One can even define real numbers (rationals and irrationals both) as
> sequences of natural numbers 0 to 9 inclusive. I thought Tom Körner had
> some notes to that effect on his web site, but I can no longer see them.

Here is another construction from the integers.

<http://arxiv.org/abs/math/0405454>

| This note describes a representation of the real numbers due to
| Schanuel. The representation lets us construct the real numbers
| from first principles. Like the well-known construction of the
| real numbers using Dedekind cuts, the idea is inspired by the
| ancient Greek theory of proportion, due to Eudoxus. However,
| unlike the Dedekind construction, the construction proceeds
| directly from the integers to the real numbers bypassing the
| intermediate construction of the rational numbers.
|
| The construction of the additive group of the reals depends on
| rather simple algebraic properties of the integers. This part of
| the construction can be generalised in several natural ways,
| e.g., with an arbitrary abelian group playing the role of the
| integers. This raises the question: what does the construction
| construct? In an appendix we sketch some generalisations and
| answer this question in some simple cases.
|
| The treatment of the main construction is intended to be
| self-contained and assumes familiarity only with elementary
| algebra in the ring of integers and with the definitions of the
| abstract algebraic notions of group, ring and field.

--
Michael Press

Date Subject Author
6/23/14 Michael Press
6/24/14 Virgil
6/24/14 Virgil
6/24/14 johngabriel2009@gmail.com
6/24/14 Virgil
6/24/14 John Gabriel
6/24/14 Michael Press
6/24/14 John Gabriel
6/24/14 Michael Press
6/27/14 Michael Press
6/27/14 John Gabriel
6/27/14 Virgil
6/24/14 Brian Q. Hutchings
6/23/14 Dan Christensen
6/23/14 Dan Christensen
6/24/14 Brian Q. Hutchings
6/30/14 ebaycaper@hotmail.com
6/30/14 Apollo
6/30/14 ebaycaper@hotmail.com
7/1/14 Virgil
7/1/14 ebaycaper@hotmail.com
7/1/14 Inverse 18 Mathematics
7/1/14 Brian Q. Hutchings
6/30/14 SPQR
6/30/14 ebaycaper@hotmail.com
6/30/14 SPQR
7/1/14 David C. Ullrich
7/1/14 SPQR
7/3/14 Brian Q. Hutchings
7/11/14 John Gabriel
7/11/14 David C. Ullrich