The following remarks reflect personal research that may be relevant to the idea of a mereological foundation.
I devised a set of sentences intended to admit a universal class to Zermelo-Fraenkel set
theory. The strategy involved a primitive part relation and a primitive membership relation with additional axioms to deal with identity and recharacterizing the part relation as a subset relation.
The proper part relation can be expressed as a self-defining predicate with a circular syntax. For this reason, I view the system as related to mereology.
The membership relation depends on the part relation, but is also introduced with a circular syntax.
The sense of these sentences is that to be a subset cannot exclude being a basic open set for a topology. To be an element cannot exclude being an element of a basic open set for a topology.
No functions or constants have such definition. A grammatical equivalence with relation to the primitive relations is defined. A first-order identity is defined after certain axioms establish familiar relations with respect to class equivalence. Second-order extensionality holds, but it is not the criterion of identity. Functions and constants may be introduced only with non-circular syntax in relation to the first-order identity predicate.
Although mereology is generally thought of in terms of the proper part relation, if one reads Lesniewski, there is a great deal of effort involved with investigation of logical equivalence. This work is done in response to Tarski's paper on primitive logistic. Tarski's analysis is done in second-order logic, as is Lesniewski's.
So, the manipulations to obtain an identity relation are consistent with Lesniewski's work, even though it does not seem that way because the usual feature discussed is the part relation.
All objects are classes, with exactly one class as a proper class. The proper part
relation is essential to establish this distinction. The first-order identity relation is also essential since the single class that is not an element of any class is unique by virtue of first-order identity. Second-order extensionality does not permit this distinction. The sole proper class is the set universe.
Again, this is consistent with Lesniewski's work. In objecting to Russell's paradox, Lesniewski develops this notion of a full class. This becomes the general mereological principle that a class and its parts are uniform.
The membership relation could be stratified using the proper part relation. But, to
establish singletons relative to the modified axiom of pairing, an empty set had to be assumed. This is not a typical mereological assumption. This stratification is comparable to what Quine found necessary in order to have a universal class for his New Foundations. If compared with Euclid, the empty set is "that which has no parts". It is the ground for units which are "that by which what exists is one".
There is a power set axiom. However, a similar axiom only collecting proper parts is included as needed to form the first-order identity. This, too, is comparable to Quine whose system has Cantorian and non-Cantorian classes. In order for the set universe to be
differentiated from its elements, proper parts had to be associated with the membership relation in the sense of a power axiom. Once a first-order identity is described, the usual power set axiom can be defined for the Cantorian "finished classes".
If these things do not sound bad enough, the model theory would necessarily be unacceptable to those committed to a predicative model construction strategy. The mereological or topological emphasis is viewed as a second-order structure in spite of the manipulations to obtain a first-order identity relation. This is consistent with the Tarskian analysis and the Lesniewskian program of research. But, it is non-standard with respect to modern foundational thinking.
In this sense, the system is Brouwerian. Logicism and logical atomism reduce the notion of object to presupposed denotations and treat the universe as Ax(x=x) with respect to ontology. When Leibniz introduced the principle of identity of indiscernibles, he did so while invoking geometric principles. The system interprets the Cantorian theory of ones in relation to his topological ideas as reflecting Leibniz' original statement. This is actually the source of the stratified membership relation. I compare it to Brouwerian ideals in that a focus on geometry is a rejection of the logicist interpretation of Leibniz principle of identity of indiscernibles.
In general, it would be best to view the structure as a closure algebra. The set universe would be the intersection over the empty set. So, the system is closed under arbitrary intersection in the same sense that an axiom of union may be interpreted as arbitrary union. With regard to statements in Aristotle, a choice has been made with regard to what
"exists". In naive set theory and set theories such as New Foundations, no distinction is made with respect to partitions in relation to negation. Aristotle remarks that one should not attempt to negate substance. A closure algebra interpretation makes a distinguishing choice of closed sets over open sets. This actually derives from the model-theoretic axiom of foundation. The transitive closures satisfy the closure axioms.
It is a very strong system. It is as least as strong as Tarski's axiom. So, it would be modeled by an inaccessible cardinal or stronger.
Although this system will never be published, it was developed carefully. I hope that these remarks help anyone who might wonder what would be involved in a mathematics based on a part relation. But, if you read Lesniewski, and the paper by Tarski, you will see that much of a Lesniewskian system has nothing to do with the part relation. The part relation had merely been an outcome of his analysis of Russell's paradox, and, he insisted that the paradox should be ignored in the development of foundations because it was the result of a mistaken analysis concerning classes.