Sets of real numbers closed under addition

Question

Recently I've had some questions about cardinality and the real numbers:

For any infinite subset $S \subseteq \mathbb R$ , can we find a set $H$ such that $S \subseteq H \subseteq \mathbb R$ , such that $H$ is closed under addition, and $|H| = |S|$ ? Can we require $H$ to be closed under multiplication as well and still have this result?

I have no experience in these sorts of problems, but I have a feeling these statements are true. Could results like this be generalized to apply to infinite subsets of an arbitrary set closed under some (possibly infinite) set of binary operations? Thanks a lot!

Answer 1

The answers to all your questions are yes. The general construction proceeds by setting $S_0 = S$ and inductively: $$S_n = \{s+t : s , t \in S_{n-1}\}$$ and letting $H = \bigcup_{i\geq 0} S_i$. It is easy to check that $H$ is closed under addition. Further, we have$$|S_n| \leq |S_{n-1}\times S_{n-1}| \leq |S_{n-1}| + \aleph_0.$$ In the case that $S_{n-1}$ is infinite, we do not need the $\aleph_0$ but you can do this construction in the finite case too. Either way, we have $|H| = |S| + \aleph_0$. This probably requires the axiom of choice or some variant.

You can generalize this to countably many $n$-ary operations where $n < \omega$ in the obvious way.

EDIT: Note that $S$ does not need to lie in some set where the binary operation exists to carry out this construction, we can carry out something like this by defining $S_n' = \{(s,t) : s,t \in S_{n-1}$ and define $s+t = (s,t)$ and quotient out by the axioms of the binary relation at each stage.