I've been looking into non-standard analysis and number systems in my own time (I'm one ofthosestudents), Surreal numbers, Hyperreal numbers, etc. Although the Surreal numbers are not a set (and therefore I am adamant they don't really exist), the Hyperreal numbers, as far as I have read, are. A totally ordered set, in fact, in which the real numbers are embedded.

But, unlike the Reals, the Hyperreals are not Dedekind-complete. If we take the set:

$\displaystyle \left\{1+\epsilon,1+2\epsilon,1+2\epsilon,\dots\ri ght\}$

It is bounded above by 2 (and 1.1, and 1.001, etc.), but has noleastupper bound. But, from what I've read, it should always be possible to "complete" a space, by Dedekind cuts. So we can define a number, call it $\displaystyle \xi$, as such:

$\displaystyle \xi=\left\{x\in \bold{H}|\exists n\in \mathbb{N} | 1+n\epsilon > x\right\}$

Now, so far I'd understood the whole process, but there is one thing that has me stumped: What are the characteristics of this number? What is $\displaystyle 2\xi$? etc. How are such things investigated? Have I constructed it properly?

Thanks in advance!