In article <email@example.com>, Peter Percival <firstname.lastname@example.org> 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.
| 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.