I currently have six blog posts on the ‘to-be-written’ pile, mostly arising out of conversations I had over the break. I had forgotten just how high-bandwidth face-time can be. Hopefully I can write most of these up before I run out of memory. Obviously most of the ideas in these posts are not due solely to me, though I’d like to think I played some role in their gestation. This post grew out of a discussion with Andy Egan, Eliza Block and Ted Sider during the drive from DC to Princeton.

It’s commonly believed that mereology entails that there could not be an even number of things. In fact, if there are finitely many things, then the number of things is 2ⁿ-1 for some positive integer n. See Antony Eagle’s good paper for a discussion of why this seems to be the case. There has (to my knowledge) been less discussion of what constraints mereology puts on the number of things if there are infinitely many things. But it isn’t obvious that it is completely innocent in this case either.

Question 1: Is it compatible with the axioms of standard mereology that there exist exactly countably many things?

It is certainly compatible with the principle that any two things have a fusion that there exist exactly countably many things. Consider a ‘Russian Doll’ world, consisting of objects o₁, o₂, etc, where for all i, j, if i is greater than j, then o_j is a proper part of o_i. Hence o_i is the fusion of itself with all the later o’s. Hence any set of o’s has a fusion – its least element. So any two things (indeed any set of things) has a fusion.

The Russian Doll world can be extended to show that it is compatible with the principle of fusions that there be exactly n things for any n. But obviously it does not satisfy all the principles of mereology. In particular, there is no object that is the mereological difference between o₁ and o₂, despite the fact that o₂ is a proper part of o₁. So I don’t know what the answer to question 1 is. I suspect the answer is no. It’s certainly impossible for an atomic world to have exactly countably many things, because if there are finitely many atoms there are finitely many things, and if there are infinitely many atoms there are uncountably many things. But I don’t know whether any worlds containing gunk could have exactly countably many things. I suspect not.

Note that I’m assuming here that it’s part of mereology that any things have a fusion. If one only assumes the much weaker principle that any two things have a fusion then it is easy to build a world with countably many things – countably many atoms and finite fusions thereof. If one assumes the slightly weaker principle that any set of things has a fusion it is still impossible for an atomic world to have exactly countably many things. But that assumption may be relevant to the next question.

Question 2: Which cardinalities are such that it is impossible (given standard mereology) that there exist exactly that many things?

I have no idea what the answer to that is. My bet is that (given a plausible version of set theory) there can’t be exactly countably many things, but assuming the continuum hypothesis that will be the only infinite cardinality such that there can’t be that many things.

Question 3: Is there any literature on this question?

If yes, please leave refs in the comments! If no, I guess someone should write up the answers to questions 1 and 2. But would anyone publish it?