Let $\mathbb{A}$ \mathbb{A^+}$ be the set of non-negative algebraic numbers. Consider the set of "polynomials" : $$\mathbb{P} = \lbrace a_0 + a_1x^{r_1} + a_2x^{r_2} + a_3x^{r_3} +\cdots + a_nx^{r_n}| a_0, a_i, r_i \in \mathbb{A}, r_i > 0, i= 1,2,\cdots,n\rbrace$$ We call $\alpha \in \mathbb{R}$ mathbb{R}, \alpha \geq 0$ extra-algebraic if there exists a polynomial in $\mathbb{P}$ satisfying $f(\alpha)=0$. Denote the set of all extra-algebraic numbers by $\mathbb{A}_E$. So, $\mathbb{A} \subset \mathbb{A}_E$.(The strict inclusion is because of numbers like $2^\sqrt2$ which are extra-algebraic but not algebraic and more by the Gelfond–Schneider theorem). We call $\beta \in \mathbb{R}$ mathbb{R}, \beta > 0$ extra-transcendental if it is not extra-algebraic. Candidates for examples of extra-transcendental numbers are $e^\pi$ and $e^\frac{-\pi}{2}$.
Question:
- Do extra-transcendental numbers exist?
- Is $\mathbb{R}/ \mathbb{R^+} - \mathbb{A}_E$ uncountable?
Many thanks.
