While thinking about my other question I found that it would be easier to talk about it if numbers just weren't so finite. There is a notion mentioned there in the comments of "list of congruences, possible or impossible" and I'm wondering if this idea has a more concise name.
So let's define a set $R$. Elements of $R$ are sequences $a_1, a_2, a_3, \ldots$ such that $\forall i. 0 \le a_i < i$ and for every finite $k$ the system
$$\left\{
\begin{array}{c}
n\equiv a_1\pmod 1 \\
n\equiv a_2\pmod 2 \\
\vdots\\
n\equiv a_k\pmod k \\
\end{array}
\right. $$
has a solution in n.
One can define the ring operations in a straightforward way, and there is an obvious injective ring homomorphism $\mathbb{Z}\to R$.
$R$ is uncountable, $R$ is compact as a subspace of Baire space $\mathbb{N^N}$.
My question is what $R$ and its elements are called.
