Nobody can even catalog the set of all subsets of positive integers closed under addition (aka subsemigroups of the infinite cyclic semigroup). See Repnitskii, Vladimir - On subsemigroup lattices without nontrivial identities. Algebra Universalis 31 (1994), no. 2, 256–265.
Update Although individually subsemigroups of $\mathbb{Z}$ are "easy" — they consist of an arithmetic progression plus a finite "garbage", the subsemigroups of $\mathbb{Q}_+$ are already very complicated. Every cancelative countable commutative semigroup without torsion (and there are lots of those) embeds into $\mathbb{R}$. That is because it embeds into a commutative countable group which, in turn, embeds into a product of copies of $\mathbb{Q}$ which is a subgroup of $\mathbb{R}$.
