TITLE: How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
QUESTION [13 upvotes]: How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?  
How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?  

I can see that results in both cases are between $\mathfrak{c}$ and $\mathfrak{2^c}$.

REPLY [9 votes]: There are $2^{\aleph_0}$ subsets of $\Bbb Q$ which are order isomorphic to $\Bbb Q$.
To see this, note that $\Bbb{Q\setminus N}$ is order isomorphic to $\Bbb Q$, and consider for every $A\subseteq\Bbb N$ the set $\Bbb{Q\setminus N}\cup A$.
Since there are no more than $2^{\aleph_0}$ countable subsets to $\Bbb R$ the answer has to be $2^{\aleph_0}$.

REPLY [2 votes]: There are continuum many countable subsets of the continuum (because $\mathfrak{c}^{\aleph_0}=2^{\aleph_0}$). Thus the answer is $\mathfrak{c}$. See this question.