# Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

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 $S$?

By the Gelfond-Schneider Theorem, $S$ includes some transcendental numbers, like $2^{\sqrt{2}}$.
• @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 Dec 26 '14 at 15:18
• I might add another question: is $S$ closed under subtraction (when the difference is positive)? – Eric Wofsey Apr 30 '16 at 1:36