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This question is about synonymy between Set theory and Mereology.

David Lewis in Mathematics is Megethology tried to reduce Set Theory to Mereology augmented with a singleton function. The following exposition is a formal capture of his Mereology.

Lets define a mereological theory $\sf M$ in mono-sorted first order logic with identity, add primitives of part-hood "$P$", and the partial unary function "$\mathfrak L $", denoting "is the lable of". Add the following axioms:

Mereological Axioms:

M1 $\textbf{Anti-symmetry: }X \ P \ Y \land Y \ P \ X \to X=Y$

M2 $\textbf{Bottom: }\exists X \forall Y: X \ P \ Y$

Define: $X = \varnothing \iff \forall Y: X \ P \ Y$

Define: $ X \ PP \ Y \iff X \ P \ Y \land \neg (Y \ P \ X)$

Define: $atom(X) \iff \forall Y: Y \ PP \ X \to Y=\varnothing$

Define: $ X\ P^*\ Y \iff atom(X) \land X \ P \ Y$

M3 $\textbf{Atomism: } \forall X (X \ P^* \ Y \to X \ P^* Z) \leftrightarrow Y \ P \ Z$

M4 $\textbf{Composition: }\\ \exists X \forall Y \ (Y \ P^* \ X \leftrightarrow Y=\varnothing \lor [atom(Y) \land \psi] ); \text{ if } X \text { is not free in } \psi $

Define: $X = \bigl[Y \mid \psi \bigr] \iff \forall Y \ (Y \ P^* \ X \leftrightarrow Y=\varnothing \lor [atom(Y) \land \psi] )$

Labeling Axioms

L1 $\textbf{Labeling: } \mathfrak L (X)=\mathfrak L (Y) \to X=Y$

L2 $\textbf{Purity: } \exists X \ (\mathfrak L(X)=Y) \leftrightarrow atom(Y) \land Y \neq \varnothing$

L3 $\textbf{Replacement: } \text { if } \psi(X,Y) \text { is a formula, then: } \\ \forall X \neg \exists^{>1} Y: \psi(X,Y), \land \\ B= \bigl[K \ P^* \ Y \mid \exists V (V=\mathfrak L(Y) \lor Y=\mathfrak L(V)) \land \exists X \ P \ A \ :\psi(X,Y)\bigr] \land \\\exists Z: \mathfrak L(A)=Z \\\to\\\exists Z: \mathfrak L(B)=Z $

L4 $\textbf{Infinity: } \exists I \exists X: \forall Y \ P^* \ X \bigl(\exists Z \ P \ X: \mathfrak L(Y)=Z \bigr) \land \mathfrak L (X)=I $

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In David Lewis account he didn't round the system with a bottom object, so a modified version of his system would be captured by $\sf M$-$\sf Bottom$ which is obtained here by removing $\sf Bottom$ axiom, re-defining atoms as objects devoid of proper parts, and stipulating the existence of a unique non-labeling atom $\varnothing$.

Is theory $\sf M$ synonymous with "$\sf MK$ sans Foundation sans Choice"?

Is $\sf M$-$\sf Bottom$ bi-interpretable with "$\sf MK$ sans Foundation sans Choice"?

My plan is to keep equality, and to identify part-hood $P$ with subset-hood relation $\subseteq$, and the labeling function $\mathfrak L$ with the singleton function $X \mapsto \{X\}$; and on the other hand to define $\in$ as: $X \in Y \iff \exists K \ P \ Y: \mathfrak L (X)=K$.

This way we come to identify classes with mereological totalities, and sets with labeled mereological totalities. All of this is following Lewis's own specifications. However, seeing that Lewis in his "Mathematics is Megethology" doesn't agree to Bottom axiom, it appears to me that without this axiom the whole Ontology would be plagued with Ur-elements, and so it appears to be so hostile to Extensionality, that I think synonymy with Set Theory cannot be achieved. However, the weaker requirement of bi-interpretability might be possible?

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In a set-theoretic context, my view is that the most compelling concept of mereology is simply the $\subseteq$ relation, and so my conception of set-theoretic mereology is simply the theory of the $\subseteq$ relation.

So this doesn't answer your question, and I won't get into your theory, which is different from how I understand things, but some readers may be interested in this alternative approach to set-theoretic mereology, and so let me mention this alternative approach, which I introduced in my paper with Makoto Kikuchi:

Namely, we study the relation $\subseteq$ in a model of set theory, and we regard this theory as "set-theoretic mereology."

It is very easy to see in ZFC set theory that $\in$ is bi-interpretable with $\subseteq$ and the singleton operator, as we note in the paper, because of the following equivalences: $$u\subseteq v\quad\iff\quad\forall x\, (x\in u\to x\in v)$$ $$y=\{x\}\quad\iff\quad \forall z\, (z\in y\iff z=x).$$ These show that from the $\in$ relation one can define both $\subseteq$ and the singleton operator. Conversely, we may define $\in$ from $\subseteq$ and singletons via $$x\in y\quad\iff\quad \{x\}\subseteq y.$$ The conclusion is that the two accounts of set theory are bi-interpretable. Set-theoretic mereology with the singleton operator is bi-interpretable with $\in$-based set theory. You can define $\in$ from $\subseteq$ and $x\mapsto\{x\}$ and conversely.

But in regard to your specific axiomatization, I wouldn't have anything more to say.

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    $\begingroup$ Are they synonymous? If we don't have a bottom object, can they be bi-interpretable? Note that we agree on using Lewis interpretations of part-hood and the singleton (labeling here) function. Your background set theory is ZFC, the axiomatic system I posed is in relation to MK-Foundation-Choice. However, still examining these questions in YOUR theory may shed a light on how they'd fair here. So, to put matters in your milieu: is ZFC synonymous with set theoretic mereology(+singleton). Next question: is ZFC bi-interpretable with a set theoretic mereology (+singleton) that lacks bottom! $\endgroup$ Commented Jan 19 at 8:03
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    $\begingroup$ I personally think the answer to the first question is to the positive. But, it is the next question that makes me wonder. Taking away bottom would almost kill Extensionality, and even then the pattern of the emerging Ur-elements might not conform with simple removal of Extensionality from ZFC, and so kills bi-interpretability. This would mean that the most natural systems of Mereology even those augmented with the singleton function, since they lack the bottom object then they won't naturally lead themselves into bi-interpretability with Set theory. Only the bottom based Mereologies can! $\endgroup$ Commented Jan 19 at 8:27
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    $\begingroup$ The point is that the bottom axiom is necessary for having a natural bi-interpretability between Mereology and Set theory, and it becomes more essential if we further demand synonymy. $\endgroup$ Commented Jan 19 at 8:37
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    $\begingroup$ Sorry, I'm just not that interested in set theory without $\emptyset$. If you want urelements, that is a separate issue, and one must add a predicate to distinguish them from the empty set, but this would happen whether you are doing mereology or element-based set theory. $\endgroup$ Commented Jan 19 at 13:39
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    $\begingroup$ Thanks! But, I'm not speaking about set theory without $\emptyset$ . I'm speaking about bi-interpretability between Mereology without Bottom (which is more natural mereologically speaking) [like the theory M-Bottom], and Set theory meaning ZFC (or MK-Found.-C), which of course has $\emptyset$ and doesn't have any Ur-elements. Notice that $\emptyset$ need not be captured as a bottom object in Mereology, for example in Lewis system we have $\emptyset$ interpreted as the mereological totality of all non-labeling atoms. $\endgroup$ Commented Jan 19 at 18:53
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There is a version of Mereology in which there is no Bottom atom, and yet it is synonymous with the above theory, and if I'm correct about the above theory $\sf M$ being synonymous with MK-Foundation-Choice, then that answers this question to the positive.

The idea is to modify the underlying mereology as to have a kind of alien atom mereologically speaking, that is an object that doesn't overlap with any other object! This was done in a prior posting. Now, to enable that here, we need to adjust composition as to forbid composing objects using that atom. Now, we use this atom to be the non-labeling atom. Here is the exposition of it:

Language: first order logic with equality, with primitives of $\subseteq$ standing for Part-hood, and $\{ * \}$ standing for the singleton function which is a partial unary function. The axioms are:

Mereological Axioms:

Define: $atom(X) \iff \forall Y \subseteq X \, (Y = X)$

M1 $\textbf{Atomism: } \forall \ atom \ X \ ( X \subseteq Y \to X \subseteq Z) \leftrightarrow Y \subseteq Z$

Define: $ X \subsetneq Y \iff X \subseteq Y \land X \neq Y $

Define: $atom^*(X) \iff atom(X) \land \exists Y: X \subsetneq Y $

M2 $\textbf{Extensionality: }\forall \ atom^* \ X \, (X \subseteq A \leftrightarrow X \subseteq B ) \to A=B$

Define: $X \ \mathcal O \ Y \iff \exists Z: Z \subseteq X \land Z \subseteq Y$

M3 $\textbf{Alien: }\exists X \forall Y: Y \neq X \to \neg X \ \mathcal O \ Y$

Define: $X = \varnothing \iff \forall Y: Y \neq X \to \neg X \ \mathcal O \ Y$

M4 $\textbf{Composition: } \exists X \forall \ atom^* \ Y \ ( Y \subseteq X \leftrightarrow \psi( Y )); \text{ if } X \text { is not free in } \psi $

Define: $X = \bigl[Y \mid \psi \bigr] \iff \forall \ atom^* \ Y \ ( Y \subseteq X \leftrightarrow \psi( Y )) $

Labeling Axioms

L1 $\textbf{Labeling: } \{X\}=\{Y\} \to X=Y$

L2 $\textbf{Purity: } \exists X \ (\{X\} =Y) \leftrightarrow atom^*(Y) $

L3 $\textbf{Replacement: } \text { if } \psi(X,Y) \text { is a formula, then: } \\ \forall X \neg \exists^{>1} Y: \psi(X,Y), \land \\ B= \bigl[ K \mid K \subseteq Y \land \exists V (V=\{Y\} \lor Y=\{V\}) \land \exists X \subseteq A \ :\psi(X,Y)\bigr] \land \\\exists Z: \{A\}=Z \\\to\\\exists Z: \{B\}=Z $

L4 $\textbf{Infinity: } \exists I \exists X: \forall \ atom \ Y \subseteq X \bigl(\exists Z \subseteq X: Z=\{Y\} \bigr) \land \{X\}=I $

Now to interpret $\sf M$ in this theory, we take the domain to be all objects here, so there is no restriction, we keep the same equality relation, we re-define $P$ as

$X \ P \ Y \equiv_{def} X \subseteq Y \lor X=\varnothing $

, and of course define $\mathfrak L$ as:

$\mathfrak L(X)=Y \equiv_{def} Y = \{X\}$

For the opposite direction, we work in $\sf M$, keep the same domain, and equality relations, re-define $\subseteq$ as:

$ X \subseteq Y \equiv_{def} X \ P \ Y \land (Y \neq \varnothing \to X \neq \varnothing) $

, and of course define the singleton function as:

$\{X\}=Y \equiv_{def} \mathfrak L(X)=Y$

Clearly the composition of these interpretations in each direction is the identity interpretation in that direction. $\small \square$

As regards the system $\sf M-Bottom$ mentioned in the question, I think it is not bi-interpretable with MK-Foundation-Choice, since it would be haunted by the same argument present in a prior posting.

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