Frank Arntzenius has a good discussion of the two models I mentioned. He also addresses the (potentially confusing) point that the models look like they’re made from atoms (this is really just an artefact of these models, that doesn’t show up in the thing being modelled.)

In which case, you are still wrong unfortunately. By Stone’s representation theorem, both the gunky models I gave you are isomorphic to a collection of subsets of some set with subsethood as the order (not the full powerset though.)

My point is still true but not really relevant, as you didn’t rely on any fusion axiom to create R\{0}. I don’t know why I thought you did, since you clearly said “union”.

“By ‘c’ I mean some fixed element (say, 0) of the set (say, R) some subsets of which we are defining parthood on, not a thing in the mereological sense.”

The subset lattice is just one of many models of mereology – all you have shown is that these kinds of models don’t have gunk.

“can’t I just take the complement of the fusion of regular open sets that don’t include zero? ”

The *union* of the regular open sets not containing 0 is R\{0}. This is not itself a regular open set so is not in the model (it’s not equal to the interior of its closure, so its not regular.) However, look at the definition of ‘fusion’ (see e.g. wikipedia) and you’ll see in this model fusions don’t correspond to unions – they correspond to the *interior of the closure* of the union. Thus the fusion you gave me is just R, the only object that doesn’t have a complement.

(By the way, {0} isn’t regular open either, so its not in the domain of the model and its not obvious you can define this fusion in the language. Obviously it doesn’t matter if you use a second order fusion axiom.)

I think the problem is I’m talking about a theory – a theory that has various models, some of which make atomlessness true, some that don’t. You appear to be talking about a particular model of that theory. If the model you are thinking of is just the power set of some set, interpreting parthood as subsethood, then its not going to make atomlessness true. Any singleton will be an atom, and every object will be a union (which *in this case* is the same as fusion) of atoms.

If you had been thinking of a different model, none of what you said would have made sense. The model might not have contained 0. Or it might have contained 0, but 0 could have had proper parts according to the model (interpret proper parthood not as subsethood but .) Just because, intuitively, an object doesn’t have any parts doesn’t mean it can’t be used to represent an object with parts in a model – you have to remember they’re formal representations.