# Is every countable well-order embeddable in \mathbb{R}? [duplicate]

Possible Duplicate:
Order types of positive reals

The title is self-explanatory.

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Yes. See the answers to this identical question - mathoverflow.net/questions/25100/order-types-of-positive-reals –  François G. Dorais Oct 29 '10 at 11:05
Thank you, and sorry for the duplicate. –  Charlie Oct 29 '10 at 11:09
The answer is yes. Choose an enumeration $\alpha_1,\alpha_2,\dots$ of your well-ordering and define a map by setting:
$$\alpha_n \mapsto \sum_{k : \alpha_k < \alpha_n} \frac1{2^k} \in \mathbb R.$$