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If we work in General Extensional Atomic Mereology [without bottom], so the primitives of the language are $P$ standing for "is a part of", and equality, now we add to it membership $\in$ relation symbol.

I observed that all rules of $\text{ZF}$ can be derived from the following Mereology-Set translation rule:

If $\phi$ is a formula in the pure language of mereology (only uses $P$ and $=$ as predicates) in which the symbol $x$ doesn't occur free, and such that all quantifiers in it are restricted to atoms (i.e. of the form $\forall x (\operatorname{atom}(x) \to\cdots)$, or $\exists x (\operatorname{atom}(x) \wedge\ldots)$, these denoted as $\forall ^* x (\cdots)$ and $\exists^* x(\ldots)$ respectively), and in which only symbols $y$, $p_1$, …, $p_n$ occur free, and if $\phi^{\in}$ is the formula obtained from $\phi$ by merely replacing each occurrence of the symbol $P$ by $\in$ and the symbols $\forall^*$, $\exists^*$ by symbols $\forall$, $\exists$ respectively, then:

$$[\forall^* p_1,\dotsc, \forall^* p_n \exists y (\operatorname{atom}(y) \wedge \forall z (\phi(z) \to z=y)] \to \forall p_1,\dotsc,\forall p_n \exists x \forall y (y \in x \leftrightarrow \phi^{\in})$$

is an axiom.

Now take the formulas: $$y=p_1,$$ $$\exists^* z \ (z \ P \ p_1 \wedge y\ P \ z),$$ $$\forall^* z \ (z \ P \ y \to z \ P \ p_1),$$ $$\exists^* x \ P \ p_1 [ \phi(x,y)], \quad\text{where}\quad\forall^* x \ P \ p_1 \ \exists^*! y (\phi(x,y)), $$ $$\forall^* I (p_1 \ P \ I \wedge \forall^* m \ P \ I (\exists^* n \ P \ I (\forall^*z \ P \ n \leftrightarrow z=m)) \to y \ P \ I) $$

All of the above formulas fulfill the hypothesis of the above schema and so the consequence would be axioms of Singletons, Set Union, Power, Replacement, and Infinity. Which are all the constructive axioms of $\text{ZF}$, except empty set, which can be simply added.

Of Note is that the known paradoxes of Cantor's, Russell's, Burali-Forti, Mirimanoff, and Lesniewski's, all are avoidable here.

I'm not really sure if that schema is consistent, but the idea is that it reflects a certain relationship between atomic Mereology and set theory, all axioms of set theory can be viewed as in some sense paralleling or rather mimicking the atomic part-hood relation of Mereology. So membership relation can be viewed as a kind of a binary relation between atoms and sets themselves are viewed as atoms.

Has such forms of mimicry between pieces of Mereology and Sets been considered before, especially along lines of setting some Mereological basis for deriving rules of set theory?

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Many people find it natural to consider the set-theoretic analogue of mereology to be the inclusion relation $\subseteq$ rather than $\in$, since after all, $\subseteq$ is reflexive and transitive and admits relative complements, and these are all axioms listed on the mereology page to which you link, but $\in$ does not have these features. (For this reason, I am a little unsure about what you are trying to do with your translation of the part-hood relation as $\in$, since this seems flawed in many ways.)

If one does proceed with $\subseteq$ as the basis of set-theoretic mereology, then there is a bit of literature.

David Lewis, for example, in his 1991 book Parts of Classes outlines an approach to mereology by which the parthood relation is understood as $\subseteq$ in a system of class theory similar to Gödel-Bernays, and Lewis's system satisfies many of the axioms on the list itemized in the article to which you link.

More recently, I've written a few articles:

Kikuchi and I undertake to analyze the nature and strength of what we call set-theoretic mereology, which is the study of the subst $\subseteq$ relation as it arises in ZFC set theory. Given a model of set theory $\langle V,\in\rangle\models\text{ZFC}$, we consider the definable reduct of this structure to the subset relation $\langle V,\subseteq\rangle$. Among other things, we prove that $\in$ is not definable from $\subseteq$ and that the theory of $\langle V,\subseteq\rangle$ is decidable — it is the theory of an unbounded atomic relatively complemented distributive lattice, and this is a finitely axiomatizable complete theory. Furthermore, we prove that all countable models of ZFC have isomorphic reducts to the subset relation. This is a sense in which set-theoretic mereology doesn't know any set theory, since it can't tell whether CH holds or fails, whether there are large cardinals or not, or ultimately, we prove, even whether there are infinite sets or not.

One philosophical point we make is that these results can be understood as an explanation of why there has been no successful implementation of set-theoretic mereology as a foundation of mathematics. Namely, one cannot use a decidable theory as a foundation of mathematics, since such a theory is not capable of serving as a foundation for the undecidable parts of mathematics, including arithmetic.

You can see more discussion of this at:

Meanwhile, let me mention that if you augment $\subseteq$ with the singleton operator $a\mapsto\{a\}$, then you can easily recover the $\in$ relation via $x\in y\leftrightarrow \{x\}\subseteq y$, and so mereology-with-singleton-operator is bi-interpretable with $\in$-based set theory.

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    $\begingroup$ Joel, this a a remarkable set of results. Collectively, they seem to suggest that mereology is so close, yet so very far from being able to provide an adequate foundation for mathematics $\endgroup$ – Philip Ehrlich Mar 15 '18 at 19:01
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    $\begingroup$ Thanks, Philip, I'm glad you like the results. I really like them, also, in part because they are mathematically interesting as well as philosophically relevant. In new work, with Ruizhi Yang, we are looking at other definable reducts of $\langle V,\in\rangle$, for example, to the unary union operator. This theory also is decidable, but if you have unary union and binary union, you can define singleton and subset and hence $\in$. It follows that binary union is not definable from unary union. $\endgroup$ – Joel David Hamkins Mar 15 '18 at 19:31
  • $\begingroup$ I read Lewis book on Parts of classes, I didn't see any real motivation for set theoretic axioms, he only explains membership using the singleton function on top of Mereology, he explains also Ur-elements, sets, classes, proper classes, etc.. but the rules of set theory he gives no real explanation of that stems from his mereology, actually in his account they are motivated by size principles, that's why he himself calls mathematics as size theory. Here you see a motivation whether direct (like the individual axioms of $\text{ZF}$ mentioned above) or indirect (like the case of replacement). $\endgroup$ – Zuhair Al-Johar Mar 15 '18 at 20:32
  • $\begingroup$ @JoelDavidHamkins most of what you are doing is to found Mereology in Set theory, or in founding set theory in some extension of Mereology (like by addition of a primitive singleton function on top of mereology). I'm doing neither here, I'm simply having both primitive relations and seeking to interpret rules of set theory by mimicry in Mereology. I didn't see that in ANY of the theories you've mentioned. $\endgroup$ – Zuhair Al-Johar Mar 23 '18 at 14:42

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