For a set $S$ of real numbers, we say that you can play sylver's coinage on it if for any infinite sequence of $x_n \in \mathbb R$, there is some $N$ such that $x_N$ is a sum of previous terms. For example, you can play on $\mathbb N$ (which corresponds to regular Sylver's coinage). Or you can play on $\mathbb Z$. You can't play on $\mathbb R$ though, since $1$, $\pi$, $\pi^2$, $\pi^3$, $\dots$ is a bad infinite sequence.
So, is there some way we can characterize which sets in $P(\mathbb R)$ you can play on.
Some examples:
- Any finite set $S$
- A subset of any playable set
- The union of two playble sets