* How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
* How many subsets of the [long line](http://en.wikipedia.org/wiki/Long_line_(topology)) $\omega\_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?


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



How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
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](https://math.stackexchange.com/questions/191006/number-of-countable-subsets-of-mathbbr).


