Wolfgang Schwarz

Sets Against Fusions

Brian points to Gabriel Uzquiano's Cardinality Puzzle about Mereology and Set Theory (PDF), which he (Gabriel) introduced a while ago in the now-deceased Philosophy from the 617 weblog. I still don't know enough set theory and mereology to competently discuss the matter, but anyway, it seems to me that perhaps the puzzle can be strengthened, as follows.

Gabriel derives a contradiction from

1) unrestricted classical mereology,

2) the correctness and all-inclusiveness of ZFCU,

3) the assumption that the individuals form a set, and

4) the assumption that there is not too much gunk.

But (4) isn't really needed if, as I believe, (the plural counterpart of) this decomposition principle holds in classical mereology:

DP) if there are k things, then there are j distinct things, with k <= 2^j;

(DP) is trivial if the k things are atoms, as then they already are distinct. If they are composed of atoms, then (DP) holds because at most 2^j things can be composed of j atoms, so the number of atoms contained in the k things satisfies the condition for j. I'm not quite sure what happens when the k things contain gunk, partly because I don't really understand gunk. But I think the argument still goes through: how could there be more than 2^j things if there is no partition of reality into at least j distinct things?

Now (2) and (3) entail that the universe is strongly inaccessible. Since inaccessible cardinals cannot be reached by taking powers, it follows by (DP) that there are just as many distinct things. But this is impossible by classical mereology, because the fusions of distinct things always outnumber those things themselves.

[Update (an hour later)] Well, if this argument worked, there would be no models of mereology with inaccessible size. But Gabriel Uzquiano says there are (on p.12 of the paper). So the argument doesn't work, and (DP) fails if there is gunk. Strange!