This theory about structures, defined as abstractions over isomorphic graphs, can interpret Set Theory in a rather creepy manner. Though the theory is largely technical, yet it is not far from being intuitive about graphs and their structure. This shows that formalization about graphs and their related structure, can constitute a more versatile formal system in which mathematics can be ingrained. In Set theory the main concepts at interplay are those of Hierarchy, order, and size. While here all of those are there, and in addition we have concepts of simplicity, continuity, circularity, and structural complexity via branching, loops, etc.. So, on the face of it, it appears to be a richer formal notion! Actually sets, numbers, shapes, etc.. all can be easily interpreted as structures of certain kind of graphs. So, in the context of this rivalry, the question is about the relative interpretability of this method in set theory.
$\frak Structure \ Theory$:
Language: first order logic
Primitives: "$=$"; "$ \varepsilon $" ; "$: \rightharpoondown $ "; "$\frak S$", read as: "is equal to" and "is a rudimentary element of", " is a directed edge from -to-", and "is the structure of". The last is a partial unary function.
Whenever "set" is mentioned here it refers to the rudimentary kind of sets presented here. While set in the sense of Set Theory, would be phrased as $\sf ZFC$ set, $\sf NF$ set, etc.. after the theory characterizing it.
The idea is to define structures as abstractions from isomorphic graphs. So, two graphs receive the same structure if and only if they are isomorphic to each other. Where the latter is a bijection between the set of nodes of each graph that preserves the edges.
I. Axioms about Equality: ID axioms.
II. Axioms about rudimentary membership:
II.a: (Extensionality) $\forall z \, (z \ \varepsilon \ x \leftrightarrow z \ \varepsilon \ y ) \to x=y$
II.b: (Elements) $\forall x \exists y: y \ \varepsilon \ x$
Define: $\operatorname {elm}(y) \iff \exists x: y \ \varepsilon \ x$
II.c: (Flatness) $x \ \varepsilon \ y \land y \neq x \to \neg \ y \ \varepsilon \ z$
II. d: (Comprehension) $ \exists y: \operatorname {elm}(y) \land \phi(y) \\ \implies \\ \exists x \forall y \, (y \ \varepsilon \ x \iff \operatorname {elm}(y) \land \phi(y))$
Define: $x=[y \mid \phi] \iff \forall y \, (y \ \varepsilon \ x \leftrightarrow \operatorname {elm}(y) \land \phi)$
Theorems: $x \ \varepsilon \ y \to x=[x] \\ x =[y] \to x=y$
Define: $\operatorname {atom}(a) \iff a=[a]$
III. Axioms about Edges:
III. a: (Directionality) $x: a \rightharpoondown b \land y: c \rightharpoondown d \implies (x=y \iff (a=c \land b=d))$
III. b: (Simplicity) $\forall x\forall a \forall b \, (x:a \rightharpoondown b \implies x,a,b \text{ are atoms})$
Define: $\operatorname {Node}(a) \iff \exists b \,\exists x \, (x: a \rightharpoondown b \lor x:b \rightharpoondown a)$
Define: $\operatorname {Edge}(x) \iff \exists a \exists b \, (x: a \rightharpoondown b)$
III.c: (Dichotomy) $\operatorname {Node}(a) \to \neg \operatorname {Edge}(a)$
III.d (Edges) $\forall \operatorname {Nodes} a,b \exists x \exists y \, (x: a \rightharpoondown b \land y:b \rightharpoondown a) $
Define Graph as a set where the nodes of every edge in it are in it.
IV. Axioms about structures
IV.a: (Domain) $\forall X \, (\exists a: a= \mathfrak S(X) \iff \operatorname {Graph}(X))$
IV.b: (Abstraction) $\forall \operatorname {Graphs} G ,F: \mathfrak S(G) = \mathfrak S(F) \iff G \sim F$
IV.c: (Structures) $\forall G : \operatorname {atom} (\mathfrak S(G)) $
IV.d (Freedum) $\forall G: \operatorname{Node}(\mathfrak S(G)) \downarrow \operatorname {Edge}(\mathfrak S(G))$
$\sim$ stands for graphical "isomorphism" defined as: $$ G \sim F \iff \exists \operatorname {bijection} f \, (f: \operatorname {Nodes}_G \to \operatorname {Nodes}_F \land\\ \forall \operatorname {Nodes} a,b \ \varepsilon \ G \, (\exists x \ \varepsilon \ G \, (x: a \rightharpoondown b) \iff \exists y \ \varepsilon \ F \, (y: f(a) \rightharpoondown f(b))))$$
Where $\operatorname {Nodes}_X$ is the set of all nodes of graph $X$.
Relations and various kinds of functions can be implemented as sets of edges under the known qualifications.
V. Axioms about Graphs
V.a: (Nodes) $\forall \operatorname {scg} G \exists a: \operatorname {Node}(a) \land \neg \ a \ \varepsilon \ G$
Where, "$\operatorname {scg}$" stands for "simple continuous graph".
Define: a set $S$ of structures of continuous graphs is said to be actuated, if and only if, there exists a discrete graph $D$ whereby for each structure $\mathfrak s \ \varepsilon \ S$ there is a single unit $d$ of $D$ such that $\mathfrak s = \mathfrak S(d)$
V.b: (Actuation) a set $S$ of structures of continuous graphs, is actuated if any of the following is met:
- Its elements can be arranged in a continuous linear manner.
- All its elements are substructures of a simple continuous graph.
- There is one-to-one correspondence with a set of nodes of a simple continuous graph.
Definitions:
- A graph is simple if no more than one edge can occur between any two nodes of it.
- Two graphs are connected if and only if a node of one graph is connected by an edge to a node of the other graph.
- Two graphs are separate if they are not connected.
- A continuous graph is one that is not the union of two separate graphs, otherwise it's discrete.
- A subgraph is a subset of a graph that is a graph.
- A unit $d$ of a discrete graph $D$ is a continuous subgraph of $D$ that is separate from $D \setminus d$.
This theory interprets $\sf ZFC$ through structures of well-founded (no infinite path), extensional (no isomorphic terminal subtrees stem from the same node), mono-rooted, accessible (no node of degree greater than the first inaccessible) trees. The main question is:
Can $\sf ZFC$ interpret this theory?
