Your definition would rule out finite sets. You don't need a bijection. A surjection from the natural numbers suffices.
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.
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.