Doubly infinite numbers

Question

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 ?

Answer 1

For a bijection between the real numbers and "doubly infinite numbers" consider the following:

Since $\mathbb R$ is equipollent with $[0,1)\times[0,1)$ we have a bijection between the two sets. Consider the pair $(a,b)$ as the inverse string of $a$ and then the string of $b$. Then we have a surjective map onto all these kind of "numbers".

As for the order relation, you can always define order relations on sets, if you want them to be somewhat useful you need to give extra constraints. However in this case there is a very natural order:

$$(a,b)\prec (c,d)\iff\begin{cases} a < c &\text{ or}\\ a=c, b<d\end{cases}$$

Do note that if you want to write $\pi$ inversely then the number you have is greater than $10^n$ for every $n\in\mathbb N$. In particular, it cannot be a real number.

Answer 2

Yes, a doubly-finite space of sequences $A^{\mathbb Z}$ with the shift acting on it is commonly seen in ergodic theory. See the topic "symbolic dynamics". Of course, specializing $A=\mathbb Z$ can be done, but this construction is interesting even when $A$ is finite. But did you mean "sequences of integers" or did you want to allow only $\{0,\dots,9\}$?