How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?

• 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 $\mathfrak{2^c}$.

• There are $\mathfrak c$ subsets of $\mathbb R$ (and of $\omega_1\times[0,1)$) of size $\mathfrak c$, so really there is only one possibility. – Andrés E. Caicedo May 26 '13 at 16:12
• @Andres Did you mean "of size $\aleph_0$"? – Hanna K. May 26 '13 at 16:30
• Note that both answers below work for the long line as well, since they're just cardinality arguments. – Noah Schweber May 26 '13 at 16:55
• (@HannaK. Yes, of course. Silly typo.) – Andrés E. Caicedo May 26 '13 at 17:35

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.
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 $2^{\aleph_0}$.