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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.

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  • $\begingroup$ You could always name them yourself (or after yourself - "the user173897 sequence) or define a function $f(S)$ being the set of $\mathbb{P}$ raised to elements of $S$ and talk in terms of functional iterates. I'm curious about whether the union of all the $A$ might contain the algebraic numbers - there's no reason it should be (or even be closed under addition and multiplication), but it would be cool. $\endgroup$ – Milo Brandt Oct 6 '14 at 22:53
  • $\begingroup$ Ha, I would not mind some eponymous sets, if they happen to be interesting! Does it (nontrivially) need to be union? I have not proven it, but I expected that $A_{n-1} \subseteq A_n \forall n \in \mathbb{N} \cup \{0\}$. But I am quite curious as to how big they are (in some sense). I would be sort of surprised to find that every algebraic number can be expressed in these terms (the limit thereof, really), because there can be weird things (summations) under radicals that I do not directly see as allowed in these sets. $\endgroup$ – user173897 Oct 6 '14 at 23:00
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    $\begingroup$ Excuse me if I'm missing something obvious, but is it clear that, say, $2^\sqrt{2}$ is even algebraic? $\endgroup$ – Ben Millwood Oct 6 '14 at 23:07
  • $\begingroup$ The set of real Euclidean constructible numbers (even in certain generalizations) seems to form a subset of most of the terms in this sequence. So, if that is truly the case (depending on the generalization), then we have that most $A_n$'s are at least as big as the set of such numbers (depending on the generalization) and is a subset (I think) of the set of all real algebraic numbers. That is a start. $\endgroup$ – user173897 Oct 6 '14 at 23:09
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    $\begingroup$ Regarding cardinality, each $A_n$ is countable, since there is an injection from finite-support sequences in $A_n$ to $A_{n+1}$, so countability follows by induction, and also the union of all $A_n$'s is countable. From this it also follows that they are all null-sets. $\endgroup$ – Mario Carneiro Jan 4 '15 at 11:46

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