I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in $\mathbb{Z}$, call it $\mathbb{P}$. Take the set of all finite products of the integer ($\mathbb{Z}$) powers of elements of $\mathbb{P}$, and unite it with $\{0\}$ and with its negative. The set produced is the set of rational numbers $\mathbb{Q}$. But let us also denote it as $A_0 = \mathbb{Q}$. Now, take the set of all finite products of the rational ($A_0$) powers of elements of $\mathbb{P}$. This is the set of the "first nice" positive algebraic numbers (I think), the ones that are basically simple radicals (in products). Let us denote it by $Ã_1 = \{\prod_{i \in \mathbb{N}}(p_i^{e_i}) < \infty: p_i \in \mathbb{P}, e_i \in A_0 \forall i \in \mathbb{N} \}$; let $A_1 = Ã_1 \cup \{0\} \cup (-Ã)$. Note that $A_0 \subset A_1$; $A_1$ is an expanded but still relatively "nice" set of numbers (not all of which are algebraic). Similarly, let us define $Ã_n$ as the set of all finite products of powers of the elements of $\mathbb{P}$ wherein the exponents belong to $A_{n-1}$; let $A_n = Ã_n \cup \{0\} \cup (-Ã_n)$.
Is there any standard name for such sets (even just $A_1$ would be nice, but I would prefer to have a name for the sequence of sets as a whole, so that I may refer to the "nth blah-blah-blah set"). What are some interesting properties of these sets? What happens as $n \rightarrow \infty$ (to which interesting, named sets does it belong)? What sorts of numbers belong to an $A_n$? Their cardinalities (I think that they possibly are countably infinite) and measures? Etc.