What is the cardinality of a bag with zero-count elements?

[dodo]

Hello, I have an aficionado curiosity, so please bear with me.

As you know, bags are sets where repeated elements are allowed. Imagine the following funny representation for a bag: instead of repeating elements, we use a set of ordered pairs, containing each distinct item plus a count of how many of its kind there are. (Suppose, for the sake of the argument, that the number of repetitions is countable.) Now the funny part: we can store items which are not in the bag by using a count of zero. The question is, what is the cardinality of such a monster?

In order to represent, say, the integers, you save each of them with a count of one. But then you also save the (rest of the) reals with a count of zero, so the cardinality of this set is at least aleph-1; but you also save elements with a zero count from sets of size aleph-2, aleph-3...

You might argue that the construction is paradoxical by design, since I am simply asking for a set which is bigger than anything you can construct. But I suspect such rejection would have an interesting consequence.

Take, from the example above, the subset of the zero-count elements, from sets of size aleph-1, -2, -3... The individual items are, in themselves, sets (Cauchy series), whose elements have, in turn, being taken from the "previous smaller set": the reals being series of elements from a set of cardinality aleph-0, and so on. So this subset is also a subset of a potentially bigger one, "the sets which do not contain themselves as a member", a phrase which should ring a bell. Would we be saying that the Russell Paradox does not exist, since it's based on a premise which is flawed in the first place?

 

[CRGreathouse]

The axiom (schema) of specification won't allow you to construct this multiset. In ZF, it doesn't exist at all. In NBG, it's a proper class (I believe).

 

[Hurkyl]

Yes, in NBG it would definitely be proper: it has an evident surjection onto the class of all things.

At least, it does if he means what it sounds like he means. e.g. that he means to represent the empty bag as

{ (x, 0) | x is a thing }​

 

[CompuChip]

I don't know much about set theory, but since we usually consider
{a, a, a, b, b, c, a, b, c, c, b} and {a, b, c} as the same set I'd say we just agree that
{ (n, 1) | n an integer} \cup { (r, 0) | r a real non-integer number } and { (n, 1) | n an integer } the same set (throwing out all elements with count 0).

But perhaps this is just too naïve.

 

[CRGreathouse]

Yes, in NBG it would definitely be proper: it has an evident surjection onto the class of all things.

It's clearly not a set. I wasn't sure if it could be constructed at all in NBG -- but I thought that it could be and that as such it would be a proper class. But you're right, limitation of size does mean that it exists.