Here’s a curious tale about sets and possible worlds: What sets there are varies between metaphysically possible worlds and for any possible world w1, the sets at w1 satisfy the full ZFC axioms and there is also a possible world w2 at which there exists a set S such that:
At w2, there is a bijection of S onto the natural numbers (i.e., a function that is one-to-one and whose range is all of the natural numbers).
The members of S are precisely the sets that exist at w1.
Suppose that this tale is true. Then assume S5 and this further principle:
- If two sets A and B are such that possibly there is a bijection between them, then they have the same numerosity.
(Here I distinguish between “numerosity” and “cardinality”: to have the same cardinality, they need to actually have a bijection.) Then:
- Necessarily, all infinite sets have the same numerosity, and in particular necessarily all infinite sets have the same numerosity as the set of natural numbers.
For if A and B are infinite sets in w1, then at w2 they are subsets of the countable-at-w2 set S, and hence at w2 they have a bijection with the naturals, and so by (3) they have the same numerosity.
Given the tale, there is then an intuitive sense in which all infinite sets are the same size. But it gets more fun than that. Add this principle:
- If two pluralities are such that possibly there is a bijection between them, then the two pluralities have the same numerosity.
(Here, a bijection between the xs and the ys is a binary relation R such that each of the xs stands in R to a unique one of the ys, and vice versa.) Then:
- Necessarily, the plurality of sets has the same numerosity as the plurality of natural numbers.
For if the xs are the plurality of sets of w1, then there will be a world w2 and a countable-at-w2 set S such that the xs are all and only the members of S. Hence, there will be a bijection between the xs and the natural numbers at w2, and hence at w1 they will have the same numerosity by (5).
So if my curious tale is true, not only does each infinite set have the same numerosity, but the plurality of sets has the same numerosity as each of these infinite sets.
We can now say that a set or plurality has countable numerosity provided that it is either finite or has the same numerosity as the naturals. Then the conclusion of the tale is that each set (finite and infinite), as well as the plurality of sets, has countable numerosity. I.e., universal countable numerosity.
But hasn’t Cantor proved this is all false? Not at all. Cantor proved that this is false if we put “cardinality” in place of “numerosity”, where cardinality is defined in terms of actual bijections while numerosity is defined in terms of possible bijections. And I think that possible bijections are a better way to get at the intuitive concept of the count of members.
Still, is my curious tale mathematically consistent? I think nobody knows. Will Brian, a colleague in the Mathematics Department, sent me a nice proof which, assuming my interpretation of its claims is correct, shows that if ZFC + “there is an inaccessible cardinal” is consistent, then so is my tale. And we have no reason to doubt that ZFC + “there is an inaccessible cardinal” is consistent. So we have no reason to doubt the consistency of the tale.As for its truth, that's a different matter. One philosophically deep question is whether there could in fact be so much variation as to what the sets are in different metaphysically possible worlds.