A lot of people find Cantor's discovery that there are different infinities paradoxical. To be honest, there are many counterintuitive things involving infinities, but this one doesn't strike me as particularly counterintuitive. Nonetheless, I want to explore the possibility that while Cantor's Theorem is of course true, it doesn't actually show that infinity comes in different sizes. Cantor's Theorem says that if A is a set, then there is no pairing (i.e., bijection) between the members of A and those of the power set PA. It follows that there are different (cardinal, but in an intuitive rather than mathematical sense) sizes of infinity given this Pairing Principle:
- PP: Two sets A and B have the same size if and only if there is a pairing between them.
A number of people have disputed the sufficiency part of PP, since it gives rise to the counterintuitive consequence that the set of primes and the set of integers have the same size as you can pair them up. But you really shouldn't both complain that there are different infinities and that PP makes the primes and the integers have the same infinite size. I am going to leave the sufficiency of PP untouched, but suggest that the pairing condition might not be necessary for sameness of size, and I will offer an alternative. That alternative seems to leave open the possibility that all infinities are the same.
To think about this, start with this thought experiment. Imagine that there is a possible world w that has some but not all of the actual world's sets, but that it still has enough sets to satisfy the ZFC axioms just as (I shall suppose) the sets of the actual world do. The set membership relation in w will be the same as in the actual world in the sense that if A is a set that exists both w and the actual world, then A has exactly the same members in both worlds (and in particular, all the actual world members of A exist in w). Then here is something that might well happen. We have two sets A and B that exist both in the actual world and in w. In the actual world, there is a pairing f between A and B. A pairing is just a set of ordered pairs satisfying some additional constraints (the first element is always from A and the second is always from B, and each element of A occurs as the first element of exactly one pair, and each element of B occurs as the second element of exactly one pair). It might, then, be the case that although A and B exist in w, f does not--it exists in the actual world but not in the impoverished world w. It might even be the case that no pairing between A and B exists in the impoverished world. In that case, we have something very interesting: A and B satisfy the pairing condition in PP in the actual world but fail to satisfy it in w. If we are to satisfy the ZFC in w, this can only happen if both A and B are infinite.
Things might go even further. We might suppose that w only contains sets that are countable in the actual world. The mathematical (much less metaphysical!) possibility of such a scenario cannot be proved from ZFC if ZFC is consistent, but it follows from the Standard Model Hypothesis which a lot of set theorists find plausible. If w only contains sets that are actually countable, then any infinite sets in w will have a pairing in the actual world. There is, thus, an important sense in which from the broader point of view of the actual world, all infinite sets in w have the same size. But w is impoverished. There are pairings that exist in the actual world but don't exist in w, and so applying PP inside w will yield the conclusion that the infinite sets in w come in different sizes. However, intuitively, it still seems true to say that these sets in w are all the same size, but w just doesn't have enough pairings to see this.
Here's one way to argue for this interpretation of the hypothesis. Plausibly:
- Pairing-Sufficiency: If there is a pairing between sets A and B, they are the same size.
- Absolute-Size: If two sets are the same size in one possible world, they are the same size in any world in which they both exist.
Thinking about the above scenario suggests a modification of PP to a Possible Pairing Principle:
- PPP: Two sets A and B have the same size if and only if possibly there is a pairing between them.
If PPP (or just (2) and (3)) is true and my w hypothesis is a genuine metaphysical possibility, then it is metaphysically possible that all infinite sets are the same size--i.e., it could be that the actual world is relevantly like w. Furthermore, we clearly don't have relevant empirical evidence to the contrary. So, if all this works, it is epistemically possible that all infinite sets are of the same size. (Of course, the most controversial part of all this is the idea that what sets there are might differ--even in the case of pure sets--between worlds.)
But perhaps this won't satisfy the people who find size differences between infinities paradoxical. For they might find it paradoxical enough that there could be infinities of different sizes, something that was definitely a part of my story (remember that I started with two worlds, one in which there were differently sized infinities and an impoverished w with all infinities of the same size according to PPP).
I think I might be able to do something to satisfy them, while at the same avoiding the biggest problem with the above story, namely the assumption that what pure sets there are differs between worlds. Here's my trick. In the above, I assumed that pairings were all sets. But in line with the Platonism suffusing all of the above arguments, let's try something. Let's allow that there are pairings that aren't sets. Those pairings would be binary relations satisfying the right formal axioms. But here I mean "relations" in the Platonic philosopher's sense, not in the mathematician's sense where a relation is a set of ordered pairs. Let's suppose, further, that corresponding to any set of ordered pairs, there is a relation which relates all and only those pairs which are found in the set. In my earlier story, I made sense of the idea that two sets in w might not have a pairing in w and yet might be the same size by adverting to pairings that exist in another world (the actual one--and then at the end I flipped things around so that w was actual). But now we do the same thing by distinguishing between mathematical pairings--namely, sets of ordered pairs satisfying the right axioms--and metaphysical pairings--namely, Platonic binary relations satisfying analogous axioms. If my earlier story is coherent (I mean that to be a weaker condition than "metaphysically possible"), then so is this one: In w, there are infinite sets that do not have a mathematical pairing, but every pair of infinite sets possibly has a metaphysical pairing. But now this story doesn't rely on varying what pure sets exist between worlds. The story appears compatible with the idea that pure sets are the same in every world. But there are, nonetheless, metaphysical pairings that do not correspond to mathematical pairings, and PPP should be interpreted with respect to the metaphysical pairings, not just the mathematical ones. Note, too, that what metaphysical pairings hold between sets might differ between possible worlds, without any variation in sets. For some of the pairings may correspond to extrinsic relations. Here is an extrinsic relation that could turn out to be a metaphysical pairing, depending on what I actually was thinking yesterday: x is related to y if and only if x and y came up in one of my thoughts yesterday in this order.
We can now suppose that this story works in every possible world. Thus, assuming the coherence of the Standard Model Hypothesis, we have a mathematically coherent story--whether the metaphysics works is another question (the story is too Platonic for my taste, and I don't share the motivation anyway)--on which (a) all infinite sets are really of the same size (and hence of the same size as the natural numbers), (b) what pure sets there are does not differ between worlds, and (c) Cantor's Theorem and all the axioms of ZF or ZFC are true. If we were to go for such a view, we would want to distinguish between sets being of the size metaphysically speaking and their having the same mathematical cardinality. The latter relationship would be defined by a version of the PP with "pairings" restricted to the mathematical ones. And then mathematics could go on as usual.