# 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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## marked as duplicate by François G. Dorais♦Oct 29 '10 at 11:04

The answer is yes. Choose an enumeration $\alpha_1,\alpha_2,\dots$ of your well-ordering and define a map by setting:
$$\alpha_n \mapsto \sum_{k : \alpha_k < \alpha_n} \frac1{2^k} \in \mathbb R.$$