|
|
|
|
|
|
by aantthony
1275 days ago
|
|
|
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 plain English, you need to construct an infinite sequence (a,b,c,d,...) which (i) consists of elements of a set S (ii) contains every element of S at least once - sometimes many times over. We also allow the members of this sequence to equal an exceptional value we denote with an asterisk: *. This is to take into account the possibility that S may be an empty set. Without this exceptional value, such a sequence cannot exist if S is empty. If S has at least 1 element, then we don't need the exceptional value.
I don't know if this means you can sensibly talk about a "next" element. The problem is that an element of S might repeat. What you're describing sounds equally like a total ordering. Assuming the Axiom Of Choice, every set has a total ordering, but the indices are not in general natural numbers, but may be ordinal numbers. You cannot in general exhibit such a total ordering, because anything proved using the Axiom Of Choice is merely known to exist, but cannot always be computed or constructed.