"Let $S$ be the set of numbers obtained from $\sqrt{2}$ and $\sqrt{3}$ by finitely many arithmetic operations ($+$, $-$, $\times$, $\div$). Show that $S$ is countable."
I can't figure out the definition of $S$. Is it true that $\{\sqrt{2}+\sqrt{3}, \sqrt{2}+\sqrt{3}\times\sqrt{2}, \sqrt{2} \div \sqrt{3}+\sqrt{3}\times\sqrt{2} \}\subseteq S$?
Yes, all of those would be in $S$. Here's a way you could define $S$ more precisely. Recursively define the set $S_n$ of numbers obtained from $\sqrt{2}$ and $\sqrt{3}$ by $n$ operations as follows. Let $S_0=\{\sqrt{2},\sqrt{3}\}$. Given $S_n$, define $S_{n+1}$ as the set of all numbers of the form $a+b,a-b,a\times b,$ or $a\div b$ for $a,b\in S_n$. Then, define $S=\bigcup_{n\in\mathbb{N}}S_n$.