I am looking for a simple way to describe the underlying set of the real numbers without getting into cauchy sequences or dedekind cuts. Furthermore, I want the description to not rely on some notion of equivalence (like how one can use the notion of coprime to give the rationals unique representatives). I think the following works:
$\mathbb{R} = \{\text{All decimal expansions | does not end in repeating 9s}\}$.
My question is am I forgetting about something or does this do the trick?
Edit: Indeed, this is not closed under the usual definition of $+$ and $\cdot$ but we can redefine these operations to “round up” when necessary. I can verify the field axioms on my own time; I am merely seeing if anyone can spot a subtlety that I missed or if my description truly does give unique representatives of the reals.
Edit 2: (Some context) The posters have given some great ways to define the reals if you want to spend a decent amount of time on it. I want to give the reals as a set of decimal expansions without any fancy notion of equivalence so that I can move on and do linear algebra. For example, if I were teaching a first year course, the field axioms would take a whole lecture and be super boring and Cauchy sequences/dekekind cuts would have students lining up to drop the course.
It will do the trick, but it will be messy and painful to prove the field axioms. Moreover, the definition in this way is somewhat arbitrary. Why choose base $10$? There is a more natural way to do it which is the Bourbaki definition of the reals. It relies on the completion construction of a uniform space. The completion of a uniform space is precisely the set of minimal Cauchy filters (so in particular, there is no equivalence involved). The reals are the completion of the rationals, and the rationals can be given the structure of a uniform space (basically, since it is topological group), thus the reals can be defined to have underlying set the set of all minimal Cauchy filters in $\mathbb Q$.
Your definition is not closed under the arithmetic operations, at least with their usual definitions on decimal expansions. For instance, take the decimal expansion
$1/9 = 0.111…$ and multiply by 9 to obtain $0.999…$, which is not an element of your set.
If you can read German, here is a detailed development of the approach Gowers is advocating at the link given in Hans Lundmark’s comment:
http://www.math.ethz.ch/~blatter/Dualbrueche_2.pdf
But note the following: Whichever approach you take, the amount of work to be done in order to verify all the details is about the same.
There are at least 3 sensible ways to do it, outlined in Spivak:
Define a real number as a Dedekind cut. A Dedekind cut consists of two sets $(L, R)$ satisfying:
Basically a Dedekind cut gets as close as it can to pinpointing a real number, without naming it (since it can’t, since we’re constructing it). I’ve personally liked this once, since it’s elementary to learn but pulls at completeness the same way a Dedekind cut pulls for a real number.
The simplest way: The real numbers are a set with operations +, -, *, / and relations < = > which follow the axioms of arithmetic, and where every nonempty set with an upper bound has a least upper bound.
Equivalence classes of Cauchy sequences are IMO the easiest way to prove that the real numbers exist, and they come kind of natural when you start talking about limits (just construct a sequence that should have the limit $2^{1/2}$ but doesn’t in the rational numbers because the number doesn’t exist).
(Going to decimals with unlimited number of digits will get you into huge trouble just defining the product of two numbers).