
> 
> **Possible Duplicate:**  
> 
> [Order types of positive reals](https://mathoverflow.net/questions/25100/order-types-of-positive-reals) 
> 
> 
> 


The title is self-explanatory.



Is every countable well-order embeddable in \mathbb{R}?
The answer is yes. Choose an enumeration $\alpha\_1,\alpha\_2,\dots$ of your well-ordering and define a map by setting:


$$\alpha\_n \mapsto \sum\_{k : \alpha\_k < \alpha\_n} \frac1{2^k} \in \mathbb R.$$


