If a set is countable, then there exists a mapping which is a one-to-one correspondence between the items and the natural numbers (that’s the definition of countable). This correspondence would always allow you to determine the “next” item (by converting to a natural, incrementing, then converting back). However there could be multiple mappings (sorted differently, for instance) and so it’s more specifically, whether it’s possible to have a next function.
In other words, there can be multiple next functions, but a set is countable iff there is at least one of them.
In other words, there can be multiple next functions, but a set is countable iff there is at least one of them.