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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 $2^{\sqrt{2}}$.

Chill2Macht
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goblin GONE
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    Seems like this set should be countable. – Jihad Dec 26 '14 at 15:07
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    @Jihad, it certainly is. That doesn't tell us much, though, since the set of algebraic numbers is also countable. – goblin GONE Dec 26 '14 at 15:08
  • Do you know any number that does not belong to this set? – Jihad Dec 26 '14 at 15:11
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    @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 $S$. – goblin GONE Dec 26 '14 at 15:18
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    I might add another question: is $S$ closed under subtraction (when the difference is positive)? – Eric Wofsey Apr 30 '16 at 01:36
  • Just a question: what tells you that this characterization determines $S$? As inclusion is not a total order... – YoTengoUnLCD Apr 30 '16 at 03:24
  • @YoTengoUnLCD, in any partially ordered set $P$, every subset $A$ of $P$ has at most one minimum element (but may have many "minimal" elements). So let $P$ denote the poset $(\mathcal{P}(\mathbb{R}_{>0}),\subseteq).$ Let $A$ denote those elements of $P$ that are closed under the relevant operations. Then $A$ has at most one least element. Regarding existence of a minimum element, it can be constructed as $\bigcap A$. – goblin GONE Apr 30 '16 at 03:29
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    S is a subset of the positive "elementary-logarithmic" or "closed-form" numbers, as defined by Chow. (http://timothychow.net/closedform.pdf) So his paper proves, assuming Schanuel's conjecture, that the positive roots of $2x^5-10x+5$ are not in S. –  Apr 03 '18 at 01:19
  • @EricWofsey, if S is closed under subtraction it would include $2^\sqrt{2}-1$, and that seems unlikely. –  Apr 03 '18 at 01:44
  • Could we build this set by transfinite recursion and then study its properties (like for the Borel Hierarchy)? – Lorenzo Jun 20 '19 at 07:26

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