Gunky ontology and virtual points

Gunk is subdivisible into smaller parts, and these are subdivisible into yet smaller parts, and this happens ad infinitum, with no smallest indivisible parts or atoms.

But here is an interesting fact: One can introduce ersatz atoms or virtual points into a gunky ontology, given some plausible mereological axioms. Suppose that O is a gunky object. Then the set M(O) of the parts of O has a partial order ≤ where x≤y if and only if x is a part of y. Now we can say that an ersatz atom of O is any ultrafilter on O with respect to the ordering.

Thus, ersatz atoms are subsets U of M(O) such that:

1. U is a non-empty proper subset of M(O)
2. if x is in U then everything that has x as a part is also in U
3. if x and y are in U, then there is a z in U such that z≤x and z≤y
4. U is maximal: any larger subset satisfying (1)-(3) is all of M(O).

We can then say that an ersatz atom U is an ersatz part of x∈M(O) provided that x∈U.

To get the existence of ersatz atoms, we need some axioms of mereology in addition to the Axiom of Choice. Fortunately, pretty weak mereological axioms suffice:

5. parthood is a partial ordering
6. O has two parts x and y that do not overlap

As usual, two things are said to overlap provided that there is something that is a part of both.

In general, given any two parts x and y that do not overlap, there will be an ersatz atom U that is an ersatz part of x but not of y. Let's further assume the strong supplementation axiom that if y is not a part of x, then there is a z that is a part of y such that z does not overlap with x. Then whenever x≠y, there will be an ersatz atom that's an ersatz part of one but not of the other. Hence, we can identify every part of O with a set of ersatz atoms. However, given gunkiness, not every set of ersatz atoms corresponds to a part. In particular, singleton sets of ersatz atoms do not correspond to parts.

So the gunk theorist can talk as if objects were made out of atoms. Now, if we have a gunky ontology, then I think we should take the parts to be non-fundamental, and grounded in the wholes rather than the other way around on pain of a grounding regress. But if we allow non-fundamental parts in our ontology, then one may worry that the gunkiness of the ontology is merely verbal and non-substantive, dependent on the verbal decision not to talk of the ersatz atoms as real parts.