On the nature of Sets

$ A \overset E {\circ\!\!-\!\! \circ} B$ is read as "$A$ and $B$ are connected given (according to; per)  $E$".

$E$ here can be described as "connectedness state", can be seen as kind of qualifier state of "being connected", so here $A$ and $B$ are among the objects that $E$ qualifies of being connected.

Axiom: $ A \overset E  {\circ\!\!-\!\! \circ} B \to B \overset E   {\circ\!\!-\!\! \circ} A$

Axiom: $ A \overset E   {\circ\!\!-\!\! \circ} B \to \operatorname {atom} (A) \land \operatorname {atom}(B)$ 

Define: $Y \ \varepsilon \ X \iff Y \ P \ X \land \operatorname {atom}(Y)$

$E$ is a uniform (continuous, homogenous) connectedness state, if and only if: $$\neg \exists A \exists B \exists C \exists D:  A \overset E   {\circ\!\!-\!\! \circ} B \land C \overset E   {\circ\!\!-\!\!\circ} D \land  B \overset E   {\circ\!\!\not -\!\! \circ} C$$; this would give the following:

$$(\exists A \exists B:  A \overset E   {\circ\!\!-\!\! \circ} B) \to \exists X:\forall Y \forall Z \, (Y  \overset E    {\circ\!\!-\!\!\circ} Z \leftrightarrow Y \ \varepsilon \ X \land Z \ \varepsilon \ X)$$

$$X \operatorname {uniformly connected}_E \leftrightarrow \forall Y \forall Z \, (Z  \overset E    {\circ\!\!-\!\!\circ} Y \leftrightarrow Y \ \varepsilon \ X \land Z \ \varepsilon \ X)$$

$$x \in E \leftrightarrow \exists X: X \operatorname {uniformly connected}_E \land x \ \varepsilon \ X$$

or simply: $X \in E \iff E \text{ is uniform connectedness state } \land \exists Y: X \overset E  {\circ\!\!-\!\!\circ} Y$

$E$ is uniform connectedness state $\to \operatorname {atom}(E)$

A clumping state is a kind of a varied (discrete; patchy) connection, so here we can have objects $A,B,D,E$ where $ A \overset {C}   {\circ\!\!-\!\! \circ} B $ and $D \overset{C}   {\circ\!\!-\!\!\circ} E$ and $ \ B \overset {C}  {\circ\!\!\not -\!\! \circ} D$, so here clearly $C$ is not uniform connectedness state, it resulted in two clumps that are separate from each other. A Clumping state seems to reflect some kind of incongruity that can hardly justify it to be an atom. An obvious example is the clumping state defined as:  $\forall X \forall Y : X \overset{C}   {\circ\!\!-\!\!\circ} Y \leftrightarrow X=Y$.

Now, ***Sets*** can be defined as uniform connectedness states, while ***Clumps*** are non-uniform connectedness state.

Since we are only interested in sets, we'd axiomatize the following to get rid of Clumps.

Axiom: $ A \overset E  {\circ\!\!-\!\! \circ} B \land C \overset E  {\circ\!\!-\!\! \circ} D \to B \overset E  {\circ\!\!-\!\! \circ} C$

So, all connectedness states are uniform!

Now, to get rid of Ur-elements we'd axiomatize:

Axiom: $ \forall X: \operatorname {atom}(X) \to X \text { is connectedness state }$

However, we'd still have non-extensional empty objects at the background, those are the mereological totalities of more than one atom, those won't be elements since they are not atoms, but they'd be empty by definition of membership. But, since they cannot be members then they won't disturb the hierarchical buildup of sets! 

**Extensionality:** $\forall C \forall E: \forall A \forall B \, (  A \overset C  {\circ\!\!-\!\! \circ} B \leftrightarrow  A \overset E  {\circ\!\!-\!\! \circ} B) \to C=E$

Actually Extensionality is natural of any connectedness state whether it is uniform or not!

So, all sets are Extensional. 

So, in some sense sets are nothing but patterns of uniform connectedness of mereological atoms. Since they are themselves mereological atoms, thus this would enable hierarchizing them up and up to whatever feasible altitudes. The net result is the establishment of a hierarchy of connectedness states, and the standard set theory can be seen to be about that hierarchy.