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 $y' = x'$. 

By transfer, every element in $^*\mathbb Q$ is a quotient of two elements from $^*\mathbb Z$. By transfer, not every element in $^*\mathbb R$ is the quotient of two elements from $^*\mathbb Z$. So, the two enlargements are not equal. 
What you are alluding to (I think) in the question is that both enlargements give rise the same completion of $\mathbb Q$, namely $\mathbb R$. 
