The Mereological grounding of set theory that follows Lewis's general outlines have various possible approaches to the definition of the empty set. But the choice of these approaches are (admited by Lewis himself) arbitrary!
Lewis preferred to denounce the class-hood of the empty set, yet maintain its precense as a starting point for building up the hierarchy of classes constituting the clay mathematics to be ingrained in. So, in his approach the empty set is all of what doesn't overlap with a class (an antity that has members), so it is what he calls as individual (a memeberless member), so its the mereological fusion of all individuals, which is itself an individual. This seems to be massive, yet it clearly leads to existence of Ur-elements which are the proper parts of the empty set itself. And if we use the unrestricted mereological composition principle, then the whole system would be plagued with Ur-elements, that there will be more of them than sets are. However, Lewis has rejected classes being parts of individuals seemingly dismissing them as Chimeras.
The other way to coin the empty set in Mereology is to actually consider it a non-labeling bottom atom. This would be the only approach that can fully grasp Lewis's aesthetic principle of parts of classes being equivalent to their subclasses, and even at the same time admitting the empty set being a class!
The problem with rounding the mereological system to have a bottom object is that: first it violates a maxim of Extensional Mereology, that is Supplementation, so it is anomalous in Mereology, and the second is that it would plague the whole system with that anomaly since the bottom object is a part of every object, leading to an omnipresent anomaly Mereologically speaking, which is clearly abhorant. This was rejected by Lewis.
I'm here asking about the following approach which is in some sense the opposite of bottom, which I personally didn't see it being mentioned by Lewis, and I'm not sure if it is discussed by other Mereologists especially in connection with Mereologically interpreting the Empty set.
The idea here is to consider the empty set as a sort of an alien object to the composition principle, i.e. we cannot compose other objects from it, a kind of a distant object, also at the same time it's to have no proper parts, and so in this sense I'll consider it as "Void", since it neither has proper parts nor is a proper part of another object, thereby not overlapping with any other object, and so being Mereologically inert. So here it is the opposite of bottom, it exists nowhere a part of itself! This doesn't violate any of the axioms of Extensional Mereology! But it does breach the Unrestricted nature of the General Sum Principle, which is by the way an arguable matter. Now this being in some sense a mild Mereological breach, then it is consistent with Mereology to put it to the minimum, so should it exists then Mereologically we are motivated to restrict it down to a single atom.
The resulting system would fully support Extensionality, and so we don't have Ur-elements, and the empty set [which is a class here] would only serve its purpose of being a start of the labeling process which further builds the hierarchy of classes we need to ground mathematics in. This way it won't have undesirable Mereological consequences, and so it quietly maintaines the purpose it is stipulated to exist for.
FORMAL WORKUP:
Language: Mono-sorted first order logic with equality "$=$", the binary relation Part-hood "$P$", the binary relation "$L$" standing for is a label of, and the constant "$\varnothing$".
Definitions & Axioms:
$Define: a \ \mathcal O \ b \iff \exists x: x \ P \ a \land x \ P \ b$
$ Define: \text{atom}(a) \iff \forall x \, (x \ P \ a \leftrightarrow x=a)$
$Define: Void(a)\iff \forall x \, (x \ \mathcal O \ a \leftrightarrow x=a)$
$Define: x \ P^* \ y \iff \text {atom}(x) \land x \ P \ y$
I: Null: $ Void(\varnothing) $
II: Atomicity: $\forall x \, \exists \text { atom } a: a \ P \ x$
III: Part-hood: $ P $ partially orders the universe:
- Reflexive: $ x \ P \ x $
- Transitive: $ x \ P \ y \ P \ z \to x \ P \ z $
- Antisymmetric: $ x \ P \ y \ P \ x \to x=x $
IV: Supplementation: $ \neg \, y \ P \ x \to \exists \text { atom } a \, ( a \ P \ y \land \neg \, a \ P \ x) $
$Define: \text { solid} (x) \iff \neg Void(x)$
V: Composition: $(\exists \text{ solid atom }x: \psi) \to \\ \exists A \forall x \, (x \ P^* \ A \leftrightarrow \text{solid atom}(x) \land \psi)$
VI: Restriction: $Void(x) \to x=\varnothing$
VII: Labeling: $\forall a \,\forall b\, \forall x\,\forall y: a \ L \ x \land b \ L \ y \to [a=b \leftrightarrow x=y]$
$Define: \text {label}(x) \iff \exists y: x \ L \ y$
VIII: Labels: $ \text {label}(x) \iff \text {solid atom}(x) $
$ Define: Class(x) \iff x=\varnothing \lor \text {solid}(x) \\ Define: y \in x \iff \exists l: l \ P \ x \land l \ L \ y \\ Define: elm(y) \iff \exists l: l \ L \ y \\Define: set(x) \iff Class(x) \land elm(x) \\ Define: y \subseteq x \iff \forall z \, (z \in y \to z \in x)$
IX: Purity: $elm(x) \land y \subseteq x \to elm(y)$
X: Reflection: $\forall sets \ \vec{p} \, ( \varphi \to \exists set \ t: \text{ trs } (t) \land \varphi^t) $
XI: Foundation: $ V $ is $\in$-well founded.
XII: Choice: $V$ is well ordered.
Where formula $\psi$ doesn't use $A$; formula $\varphi$ has all its parameters among symbols $\vec{p}$, doesn't use $t$, use only $\in, =$ as predicates, but allowed to use definable function term symbols; $\varphi^t$ is obtained merely from bounding all quantifiers in $\varphi$ by "$\subseteq t$"; $\text {trs}$ stands for is $\in$-transitive, i.e. closure under $\in$.
Note: a definable function term symbol is a term symbol of the form $f(x_1,..,x_n)$ for $n=0,1,2,...$, where $f$ is a function symbol that is defined completely in set theoretic formulation, i.e. all parameters of the defining formula (which includes among them all $x_1,..,x_n$ symbols) are sets, and all quantifiers in it are restricted by the predicate $set$, only use $=,\in$ as predicates, and contain no defined symbol (other than $\in$).
The above system inteprets all axioms of ZFC.
Is there a clear inconsistency with the above system?
Is this interpretation of the empty set Mereologically plausible?
