"Pure Maths"- Yuck. I prefer Applied Maths.
But ThinkABit you made it interesting. Kudos.
I don't mind pure maths in small doses :)

Good comments from everyone here. Kudos.

thinkabit,
How do you get through the day with these things on your mind or are you a hard drive ? :-)

A real number is defined as a non-empty set, S, of rational numbers, such that if:

1) x and y are rational numbers¹; AND
2) x is in S; AND
3) y is less than² x,

then y is also in S.

http://en.wikipedia.org/wiki/Dedekind_cut

A rational number is converted into a real number by mapping it to the set of all smaller rational numbers, excluding itself.

Both the sets described by "1" (or "1/1") and "0.999999..." contain exactly the same members³, hence they are the same.

---
¹ i.e. they can each be expressed as a fraction of two integers, "a/b" and "c/d" respectively
² i.e. c×b < a×d
³ i.e. all the rational numbers "u/v" such that 'u' and 'v' are integers, v > 0 and u < v

Yuyutsu,
Trust you to throw in your 2xBob. :-)

I just re-read what I posted previously today and should correct something: Where is asked "So just how big is this set?" I should really be asking "So how complicated is this set?"

Yuyutsu:
The equivalence class of Cauchy sequences approaching the same limit is the standard construction method taught these days (atleast in Austrlia a few decades ago when I was studying this stuff). Dedekind cuts were certainly a popular construction over a hundred years ago. However I suspect that it fell out of favour as since students were studying more and more rigorous versions of calculus at an early stage where they were exposed to the technical concepts of detla-eplison limits and then when for those that moved on to studying the deeper concepts of the real number line, such as its construction, the professors could leverage this established knowledge base by constructing the reals with Cauchy sequences. Also, I believe a lot of the constructions of the commonly used examples, such as square root of 2, is easier with Cauchy sequences then with Dekekind- but I don't know this for certain since I've never really used Dedekind cuts, I've just have a rough idea how they work.

By the way- In your Dedekind cut definition you've covered the "closed downward" requirement but haven't you missed "no greatest element" requirement?

Also, if you are claiming the integer 1 and rational 1/1 and real 0.9999.. are the same set then this is wrong! (I'm not exactly clear what your claiming in that line.) However, it is true that the real number that the rational 1/1 homomorphically maps to is the same real represented by 0.999... (Is this what you are saying?)