Consider the following theorem of Cantor:

If one has a simply-ordered set (M.<) which fulfills the three conditions: a) |M| = aleph-null; b) M has no greatest or least element; c) M is everywhere dense; then M has the order-type of the rationals (quoted from Dauben's book GEORG CANTOR,pg 187).

Let (M,<) be a simply-ordered set of points satisfying a,b,and c on which the automorphism

group defined on M acts transitively by translation so that all elements m of M 'look alike'.

I want to consider a completion of (M,<) by 'Dedekind Cuts' C of M to a simply-ordered set

(R,<) which has the following properties:

a') (R,<) is perfect; b') (R,<) contains the set (M.<) previously defined which is so related to (R,<) that between any two elements r(0), r(1) of R, elements of M occur (by another theorem of Cantor, (Dauben, ibid, pg 191) (R,<) has the order-type of the reals--sorry for the possible redundancy here, I hope it is not too annoying).

Given that (M,<) is defined as a set of points on which the automorphism group defined

on M acts transitively by translation (so that all elements m of M 'look alike'), is

the completion (R,<) of (M,<) satisfying a',b' the unique completion and does |R|=

aleph-one? (Again, I am sorry if the question has possible hidden vagueness--people

who might try to answer this question can suggest how to remove these.)