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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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up vote 5 down vote accepted

The answer is yes. Choose an enumeration $\alpha_1,\alpha_2,\dots$ of your well-ordering and define a map by setting:

$$\alpha_n \mapsto \sum_{k : \alpha_k < \alpha_n} \frac1{2^k} \in \mathbb R.$$

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