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Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$

Questions:

1. Do either $S$ or its elements have an accepted name?

2. Where can I learn more about the set $S$ and it elements? (Reference Request)

3. Do there exist positive, real algebraic numbers which are not in $S$?

4. Are either $e$ or $\pi$ elements of $S$?

What I Already Know:

By the Gelfond-Schneider Theorem, $S$ includes some transcendental numbers, like $2^{\sqrt{2}}$.

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  • $\begingroup$ Seems like this set should be countable. $\endgroup$ – Jihad Dec 26 '14 at 15:07
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    $\begingroup$ @Jihad, it certainly is. That doesn't tell us much, though, since the set of algebraic numbers is also countable. $\endgroup$ – goblin Dec 26 '14 at 15:08
  • $\begingroup$ Do you know any number that does not belong to this set? $\endgroup$ – Jihad Dec 26 '14 at 15:11
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    $\begingroup$ @Jihad, I suppose that any incomputable would not belong to $S,$ and there exist incomputable numbers that can be defined explicitly. Although my knowledge of computability theory is sufficiently poor that I am not 100% certain of this statement. More interestingly perhaps, I have hunch that $e$ and $\pi$ do not belong to $S$, and that there might exist a fifth-degree polynomial (or higher) over $\mathbb{R}$ with a solution that does not belong to $S$. $\endgroup$ – goblin Dec 26 '14 at 15:18
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    $\begingroup$ I might add another question: is $S$ closed under subtraction (when the difference is positive)? $\endgroup$ – Eric Wofsey Apr 30 '16 at 1:36

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