The main difference with this formal theory is that it depends in an essential manner on a defined notion of class, set, and set membership $\in$, rather than the usual appraoch of leaving them undefined.
This theory is an extension of Atomic General Extensional Mereology with Bottom $\sf AGEM+Bottom$, by adding a primitive binary relation $L$ standing for labeling, so $k \, L \, x$ is read as "$k$ labels $x$", or can be read as "$k$ is a label of $x$". The axioms about Labeling are:
Labels: $a \, L \, x \to atom(a)$
"All labels are atoms"
Labeling: $a \, L \, x \land a \, L \, y \to x=y$
"No label labels two distinct objects"
Empty: $\forall \text{ atom } a \, (\not \exists x \,( a \, L \, x) \iff a=\varnothing)$
Where $\varnothing$ is the Bottom atom.
"There is only one non-labeling atom which is the Bottom object".
Define: $class(x) \iff \forall a \, \forall b \, \forall y \, (a \, P \, x \land a \, L \, y \land b \, L \, y \to b \, P \, x)$
"A class is an equivalence totality of labels under equal labeling"
Define: $y \in x \iff class(x) \land \exists a \, P \, x : a \, L \, y$
An element is what's labeled by an atom of a class.
Define: $set(x) \iff class(x) \land \exists y : x \in y$
A set is a class that is an element of a class
Define: $pure(x) \iff class(x) \land \forall a \, P \, x \, \forall b \, \forall y \, (a \, L \, y \land b \, L \, y \to a=b) $
A pure class is a class of unique labels.
From the above we get the following aesthetic Lemma:
Lemma: $\forall x \, (pure(x) \iff \forall y \, (y \, P \, x \iff y \subseteq x))$
Where $y \subseteq x \iff \forall z \in y \, (z \in x)$
That is, "pure classes are those whose parts are their subclasses and vice verse".
I call such classes as Lewisian, after David Lewis, who introduced those classes in his Parts of Classes.
Define: $pureset(x) \iff pure(x) \land \exists! \, a: a \, L \, x$
So a pureset is not just a pure class that is a set, in addition to that it must be uniquely labeled! Those I also call as "Lewisian sets".
Reflection: if $\phi$ is a formula of the language of set theory, whose parameters are among $w_1,..,w_n$
$n=0,1,2,...\\\forall \, puresets \, w_1,..,w_n: \\\forall x \, ( \phi \to pure(x)) \\\to \\ \forall x \, ( \phi \to \exists! \, a: a \, L \, x)$
Where the langauge of set theory includes formulations only using $\in;=$ as predicates. (Note that when $n=0$, the axiom becomes parameterless).
In English: All predicates definable in the language of set theory from pureset parameters if they only hold of pure classes, then all of those classes are pure sets
The suprising thing to me is that this attempt have a simply stated single schema on top of the axioms of Labeled Mereology, that can interpet all of Ackermann's set theory and therefore interpret ZFC. Simply take Ackermann's mystereous set predicate to be hereditarily pureset and all axioms of his set theory would be go through under that interpretation.
The motivation for this axiom is maximally turning pure classes into pure sets, and thereby constructing a strong buildup that is made from pure sets that enables strong kinds of comprehension over them. Now I'm not sure how much this is justified philosophically, but definitely it inspires strong technical developments in set theory, that enabled having a theory as strong as ZFC, and yet opens the door wide for it to be extended by setting axioms for the rest of kinds of sets.
The point here is that this axiom does depend in an essential manner on the defined notions of sets, classes and their membership; it essentially depends on the internal make-up of sets, I mean in Mereological terms! That is, what they are composed of. So it features Ackermann's sub-world $V$ as the world of those precious kind of sets, the most pure form that a set can be of, i.e. those sets which were described by Lewis. On the other hands proper classes are classes that are not Lewisian-sets. This is a tempting explanation of Ackermann's set world.
Two questions; the first is mathematical, the other is philosophical:
Is this theory interpretable in Ackermann's set theory, or in ZFC?
A part from the question of legitmacy of having a Bottom atom in Mereology (which can be done without it here, but the formalization becomes more complex), did the definitions given here adequatly capture the true nature of the notions they define? I mean as relates to those notions as treated in modern standard formal set theories which usually do not use Ur-elements.
