$ \newcommand{\Pt}{ \ \mathbb P \ }
\newcommand {\cz}{\ C_z \ }
\newcommand {\eps}{\ \varepsilon \ }$**Logic:** first order logic with equality

**Extra-logical primitives:** 

 - "$\varepsilon$" standing for the binary relation "is an atom part of", 
 - a ternary relation symbol $C_z$ standing for "is connected to according to $z$".  

Axioms:

 1. **Extensionality:** $\forall z \, ( z \eps x \leftrightarrow z \eps y ) \to x=y$.
 2. **Atomicity:** $\forall x \exists y: y \eps x$.
 3. **Rudimentary Membership:** $\forall y \forall x: y \eps x \to \forall z \, (z \eps y \to z=y)$.
 4. **Comprehension:** $\exists y \, ( y \eps y \land \phi(y)) \to \exists x \forall y \, (y \eps x \leftrightarrow y \eps y \land \phi(y))$.

 Now, we can retrieve all of the known Mereological axioms about the part-hood relation $\mathbb P$ by simply defining parthood as: $$ x \Pt y \iff \forall z \eps x (z \eps y).$$
It can be easily verified that $\varepsilon$ is the atomic part-hood with respect to $\mathbb P$. We do have: $$\forall x: x \eps x \iff \forall y: y \Pt x \to y=x.$$

 5. **Atomhood:** $x \cz y \to  x \eps x \land  y\eps y \land z\eps z$.

      **Define:** $\operatorname{state}(z) \iff \exists x \exists y : x \cz y$.
   
 6.  **Reflexive:** $\operatorname{state}(z) \land x \eps x \to x \cz x $.
    
   
 7.  **Symmetric:** $x \cz y \to y \cz x$.
  

      

 

 8. **States Extensionality:**
   $\forall \operatorname {states}z,k: \forall x \forall y \, (x \cz y \leftrightarrow x \ C_k \ y) \to z=k
    $.
     
     **Define:** $x \in z \iff \operatorname{state}(z) \land x \eps x \, \land \, \forall y: y \neq x \to \neg y \cz x $.

     **Define:** $\operatorname {contacting}_z(x) \iff \exists y: y \neq x \land x \cz y$.

     **Define:** $\operatorname {uniform}(z)  \iff \forall x \forall y \, ( \operatorname{contacting}_z(x) \land \operatorname{contacting}_z(y)  \to x \cz y)$.


    **Define:** $\operatorname {set} (z) \iff  \operatorname {state}(z) \land  \operatorname {uniform} (z)$.

     **Define:** $\operatorname {nse}(z) \iff z \eps z \land \neg \operatorname {set}(z)  $.

    **Define:** $\operatorname {ur}(z) \iff \operatorname {nse}(z) \land \neg \exists x: x \in z $.
 
/ 

"$\operatorname {nse}$" is short for non-set element.
Now, I think the above system is consistent. What it lacks is a comprehension axiom for relation $C$. I'm contemplating the following one:

**Connections:** if $\phi(x,y)$ is formula that doesn't use the symbols "$C$" and "$z$", whose free variables are "$x,y,w_1,\dotsc,w_n$"; then:

$$w_1 \eps w_1 \land  \cdots \land w_n \eps w_n \land \forall x \forall y [\phi(x,y) \to \phi(y,x)] \\ \to \\  \exists \operatorname {state} z: \forall x \forall y \,  (x \cz y  \land x \neq y \leftrightarrow x \neq y \land  x \eps x \land y \eps y \land \phi(x,y) ).$$

It might be possible to allow $C$ in $\phi$ but only as $C_{w_i}$. 

I think if the above theory is consistent then it could possibly be equal to second order arithmetic? One peculiar matter is that it does have a universal set. And, this is a suspicious feature.

> Is that theory consistent?

> If we allow $C$ in Connections as mentioned above would this be consistent?