I'm having a problem with your remark: "A set is denumerable (contains a denumerable amount of members) if its members can be mapped by some rule or rules, one to one, with the positive integers, and that set can contain either a finite or infinite amount of members." I guess that I don't know what the referent of "that set", in the last clause, is. There seem to be only two candidates: (a) a denumerable set, or (b) the positive integers. But neither of those sets "can contain either a finite or infinite amount of members." They both must contain infinitely many members; neither can contain finitely many members, though you clearly seem to say that at least one of them can.
Perhaps you were thrown off by something else--my claim that there are infinite sets that are not denumerable...sets that are larger than the sets of natural numbers and positive integers. Are you comfortable with that, or might that have drawn you astray?