
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 wellknown 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  selfcontained 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

