The naturals, integers, rationals, algebraic, computable numbers are all countable subsets of the reals. What are some more interesting esoteric subsets of the reals which are countable? Which of those countable subsets actually form a field?
[Math] Countable subsets of the reals
real numbers
Related Solutions
For the basic notions of the calculus, like continuity and limits, you don't need the reals if you are happy to substitute them for something abstract. There are two ways this can be done. One is topology, but this is almost certainly not going to appeal to someone who did not already know enough calculus. The axiomatics of topology allows you to speak rigorously of the basic notion of calculus without mentioning the reals. Another possibility is to generalize metric spaces. Classically a metric space takes values in the reals, but you can replace the reals by what is called a value quantale. This axiomatization is much more easy to digest, so it can be used to introduce metric spaces without the reals, and again introduce the common notion of the calculus.
You are what is the minimal extension of the rationals needed to speak of calculus. Well, it would seem that a crucial property to have is that whatever the extension is it must be a complete lattice. Any complete lattice extension of the rationals must contain the reals, so the minimal such would be the reals.
If we pick a particular description system, it seems naively that the set of real numbers that will be describable is countable. But, at the same time, it will seem that by referring to that description system itself we can describe even more real numbers that were not describable in the original system. Indeed, given any sequence of real numbers, the proof of the Baire category theorem allows us to concretely describe a real number that was not in the original sequence.
The topic of describable or definable real numbers is full of technical difficulties that can appear paradoxical. For example, in some models of ZFC every real number is definable, and at the same time that model (like all models of ZFC) believes its set of reals is uncountable. See https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb for one more thorough explanation.
As a particular example of a subtle technical difficulty, the question reads
Of course, that the real numbers have no "holes" is nice, but we can never actually hit a hole, as by hitting we would have to describe it.".
This argument reverses the roles of the construction and the description system. If we begin with a fixed description system, there is no reason to think that every real number "hole" we can concretely construct as a limit of describable numbers will be describable by that particular description system. The real number "hole" may only be describable by some other description system. That leads us to look at unending sequences of description systems, which changes the setting significantly, because we are no longer expecting every number to be describable by the same system.
Best Answer
The rational and (real) algebraic numbers are of course fields which are contained in $\mathbb{R}$. The computable numbers are also a field, to see this, you'd have to show that the sum, product, difference and quotient (whenever this makes sense) of any two computable numbers is computable. Also, you might want to observe that the computable numbers contain the (real) algebraic numbers as a subset. If you want some more, take any countable field you like and adjoin a countable number of new elements, and you will get a countable field.
I'm not quite sure how to answer the first part of your question. What do you mean by interesting? Do you want some examples of sets which have some mathematical utility, and turn out to be countable? For some interesting non-examples you can show that any non-empty open subset of $\mathbb{R}$ has size continuum, and any closed subset of $\mathbb{R}$ is either countable (or finite) or of size continuum.