FLAT SET THEORY
Here the theory is about Flat sets, it has Mereological underpinnings, a set is contemplated as a collection of atoms (i.e. objects having no proper parts, which are embodied here by singletons that are members of themselves, generally known as Quine atoms), so we have only one tier of multipleton sets! Hence the term "flat". Here the underlying idea is of sets being exactly their members, so a set is its members in their specified forms and nothing else, and so a set cannot be composed of material that is extra to what its members are composed of, neither it can take a form other than the forms of its members, so a set is a specific material arranged in a specific form, and it is exactly the totality of material and forms of its elements, and there is nothing more into sets than that! So there cannot be an empty set, and there cannot be singleton sets that are distinct from their sole elements! However, to make this approach powerful enough to capture mathematics, then we'd need to have a way of implementing relations in it. The best and most simple way is to add directional links i.e. atoms that link atoms to atoms in one direction. There is nothing in the environs of Mereology to oppose that. Truly Mereology has a problem with a general concept of ordered pair, like those occurring between multipleton sets, which is quite hostile to Mereology, but as regards atoms, there is no hostility whatsoever. Accordingly extending Mereology with directional linking atoms that directionally link atoms to atoms could be considered as Mereology friendly, thereby constituting a genuine extension to Mereology! Adding those would confer the necessary tools to coin strong Flat set theories, that I'd call as Relational Flate Set theories.
Relational Flat Set Theory:
Language: First order logic with equality ‘‘=", membership ‘‘∈", and directional linking L, the latter is a partial two place function symbol,
Where l=L(a,b) is read as l links a to b.
Axioms:
Extensionality ∀(z∈x↔z∈y)→x=y
Membership: x∈y→x={x}
Actuality: ∀x∃y(y∈x)
Composition: ∃x(Sg(x)ϕ)→∃s∀x(x∈s↔Sg(x)ϕ)
Multiplicity: ∃>3x:Sg(x)
Naturality: l=L(a,b)→l,a,b are singletons
Links: ∀ singletons a,b∃l:l=L(a,b)
Direction: L(a,b)=L(c,d)↔(a=c∧b=d)
Well Ordering: ∃R:R well orders the set of all singletons
Where ‘‘Sg" means "is a singleton".
Or instead of axioms of Multiplicity and Well ordering, one can add only one axiom of infinity stating the existence of a well ordered infinite set, where an infinite set can be defined after Dedekind.
Now the above is strong! it interprets second order arithmetic in which most of traditional mathematics can be formulated.
Not only that! one can easily extend this method depending on rather elementary notions related to size and order as to reach the strength of ZFCor even go beyond it.
The point is that Relational Flat Set Theory is also strong! And naive conceptions about it is nearly as strong as standard set theory itself. This means that Mereology which is the basis for this kind of set theory, is strong! And it indeed can serve as a foundation for mathematics. And it might even offer a more suitable alternative to formalize some notions in applications that require the kind of flatness presented here, what could be labeled as "hierarchy-hostile" notions. I'm not sure if there are such notions in applications, but if so, then the above might be the suitable formal arena in which they can thrive.
See my answers to:
FLAT SET THEORY WITH SET CODES:
Language: First order logic with equality ‘‘="; membership ‘‘∈"; the binary relation <; and the one place total function symbol set.
Axioms:
Extensionality ∀(z∈x↔z∈y)→x=y
Membership: x∈y→x={x}
Actuality: ∀x∃y(y∈x)
Composition: ∃x(∃!y(y∈x)ϕ)→∃s∀x(x∈s↔∃!y(y∈x)ϕ)
Let SS be the set of all singletons.
Well ordering:< is a well ordering on S.
Infinity: ∃x[∃y(y<x)∧∀y<x∃z(y<z<x)]
Coding: ∀x∀y:set(x)=set(y)→x=y
Codes: ∀x [∃l(x ⊆{z:z≤l}) ↔ ∃y∈S (set(x)=y)]
Initiation: ∃x∈S:∄y [x=set(y)]
Define: |A|=x⟺x=min k:∃f(f:A↣{m:m≤k})
Size: |x|=κ→∃l∀y∈x (l>y)
Successor cardinals: ∀x∈S∃y(|y|>x)
The theory minus the last two axioms can interpret Kripke-Plateq set theory, and thus can define L within it, adding the last two axioms would serve in inflate that L to that of ZF + |V|=|L|, thus interpreting the latter.
So coding would easily translate the flat sets into the hierarchical sets of ZFC.
Another simpler way to present flat set theory is:
Language: First order logic with equality ‘‘="; membership ‘‘∈"‘; and the one place partial function symbol "set".
Axioms:
Extensionality ∀(z∈x↔z∈y)→x=y
Membership: x∈y→x={x}
Actuality: ∀x∃y(y∈x)
Composition: ∃x(∃!y(y∈x)ϕ)→∃s∀x(x∈s↔∃!y(y∈x)ϕ)
Let S be the set of all singletons.
Multiplicity: ∃x,y∈S (x≠y)
Coding: ∀x∀y:set(x)=set(y)→x=y
Codes: ∀a,b∈S∃x∈S: x=set ({a,b})
Well ordering: ∃R:R is a well ordering on S