What is cardinality of a set of all abelian operations on $\mathbb{R}$ up to an isomorphism?

Question

Let $A$ be the set of all operations $+$ such that $(\mathbb{R},+)$ is abelian group. I will define relation $\sim$ on $A$ in a following way: $+_1\sim+_2, ~+_1, +_2\in A$ if and only if $(\mathbb{R},+_1)$ and $(\mathbb{R},+_2)$ are isomorphic.
My question is then:
What is cardinality of $A/\sim$?

I know that $A/\sim$ has at least two elements, but I suppose there are a lot more of them. I have little experience with group theory and I don't have any idea how to solve this problem.

Answer 1

The cardinality is $2^{|\mathbb{R}|}$ (i.e., $2^{2^{\aleph_0}}$).

First, any group operation on $\mathbb{R}$ is a function from $\mathbb{R}^2$ to $\mathbb{R}$, so it can be identified with a subset of $\mathbb{R}^3$ (namely, the graph of the function). As there are $2^{2^{\aleph_0}}$ subsets of $\mathbb{R}^3$, that's an upper bound on the number of abelian group operations on $\mathbb{R}$.

I don't know a simple way to show that this upper bound is attained (but I hope someone else does, and will post it!). Here's one way to get that result (but it is almost surely overkill): The complete first-order theory of the group $(\mathbb{Z}^\omega, +)$ is not superstable, and therefore (by a theorem of Shelah) has $2^{2^{\aleph_0}}$ non-isomorphic models of cardinality $2^{\aleph_0}$. Each of these models can be realized by a group operation on the set $\mathbb{R}$ (by "transport of structure"). In particular, there are $2^{2^{\aleph_0}}$ non-isomorphic abelian group operations on the set $\mathbb{R}$.