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Real numbers are sequences of integers which are infinite in one direction. If I have a string which is infinite in both directions, say ...345123985..., then I can form an injection from these strings to R^2 and then into R (just pick a point and read of a real number for both direction), is there any simple surjection from R to these doubly infinite strings?

Is there any context in which these numbers arise or have any use?

I consider two strings equal if they are equal after a translation. Is there any way to define an order-relation on them ?

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    The simplest surjection I see from $\mathbb{R}$ to your strings is to make $\mathbb{R}\Longleftrightarrow (0,1)$, then read the odd number digits to form one string infinite in one direction, read the even digits to form another, and concatenate them with the second reversed. You have to worry about terminating decimals, but those are countable.2011-10-09

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