Given a countable set on which a total ordering is defined, it is not always possible to list the full set in ascending order. The set of rationals is a well-known example.
Show that it is impossible to list the rational numbers in increasing order
But are there conditions under which such a listing is possible? Ideally, the conditions would be both necessary and sufficient conditions, but I suppose weak sufficient conditions will, eh, suffice. For instance, can the support of a discrete random variable on a probability space always be listed in ascending order?
Let S be a countable linear order.
Assume for all x,y in S, (x,y) is not order dense.
That is one requirement.
Another requirement is to prevent intervals of the form
(-1/2, -1/3,.. -1/n,.. ..1/n,.. 1/3, 1/2).