Is ${}^{*}\mathbb{Q}={}^{*}\mathbb{R}$? This is a follow-up question of a previous one.
Is it true ${}^{*}\mathbb{Q}={}^{*}\mathbb{R}$ ?
It seems to me it's true because for any $x \in{}^{*}\mathbb{R}$, there is a $y \in {}^{*}\mathbb{Q}$ such that $y < x$ and vice versa, and because for any $x' \in \mathbb{R}$, there is a $y' \in {}^{*}\mathbb{Q}$ such that $y' = x'$. 

No, ${}^*\Bbb Q\ne{}^*\Bbb R$. It’s not true that $\Bbb R\subseteq{}^*\Bbb Q$. In the ultrapower construction this is especially easy to see: in ${}^*\Bbb R$, $\Bbb R$ is identified with the set of equivalence classes of constant sequences of reals, and the class of a constant sequence with irrational value is easily seen to be different from every element of ${}^*\Bbb Q$, though it’s also easy to see that there are elements of ${}^*\Bbb Q$ infinitely close to each such class.
