It is well-known that the set of rational numbers $ \mathrm Q$ is dense; that is, given any two rationals, say $ r\ne s,$ then there exist infinitely many rationals $ r_i$ such that $ r<r_i<s$ for every integer $ i.$

Obviously, this property depends on the usual ordering of the rationals; this can also be seen by using Cantor’s famous ordering in order to show the equinumerousity of the integers $ \mathrm Z$ and $ \mathrm Q,$ beginning $ $ \frac 01,\frac 11,\frac 12,\frac 21,\frac 13,\frac 31,\cdots.$ $ Clearly, *each* rational is in this sequence, so that there is no rational between each pair, the first two, say.

My question is about the possibility of a reverse process. Is there a way to order the ** integers** so that they become dense? Clearly one will have to design an ordering different from the usual one to make this happen, were it possible. But I don’t even have a hint where to start, or whether it is in fact possible.

Thank you.