Countable subsets of the reals

Question

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?

Answer 1

-Take any countable set $\mathcal{F} = \bigsqcup \limits_{p \in \mathbb{N}^*} \mathcal{F}_p$ of functions (or partial functions) on $\mathbb{R}$ ($\mathcal{F}_p$ is the subset of functions with $p$ arguments), now define by induction a sequence of countable subfields of $\mathbb{R}$ by $F_0:=\mathbb{Q}$, and $F_{n+1}$ is the subfield generated by $\{ x \ | \ \exists p \in \mathbb{N}, \exists f \in \mathcal{F}_p, \exists x_1,...,x_p \in F_n, x = f(x_1,...,x_p)\}$. Define $F:= \bigcup \limits_{n \in \mathbb{N}} F_n$. Then $F$ is the smallest subfield of $\mathbb{R}$ stable by functions in $\mathcal{F}$.

For instance if $\mathcal{F} = \{\exp\}$, you get the smallest exponential subfield of $\mathbb{R}$, if $\mathcal{F} =\{\exp;\ln\}$, then it is the smallest exponential-logarithmic subfield of $\mathbb{R}$.

-If a subfield $F$ of $\mathbb{R}$ is countable, then some of its universal closure with respect to some conditions are countable: its euclidean closure, its real closure...

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So basically, since countability (like any infinite cardinallity) is preserved by countable unions, many constructions based on a countable set produce a countable set with the desired properties.

Answer 2

A more-recently proposed, interesting, countable set: the periods as explained in

Kontsevich, Maxim & Zagier, Don
"Periods"
in: Mathematics unlimited—2001 and beyond, pp. 771–808, Springer, Berlin, 2001.

It is (conjecturally) not a field, since it contains $\pi$ but (conjecturally) not $1/\pi$.

Answer 3

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.

Answer 4

These subsets are subgroups of the additive group of real numbers. If we consider the cosets of each of these subgroups, they are all countable sub sets of $\mathbb R$, but not fields (excluding the coset of rationals & computable numbers).