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Non-links: $\exists x: \neg \exists a \exists b: a \overset{x} - b$

Non-links: $\exists x: atom(x) \land \neg \exists a \exists b: a \overset{x} - b$

C3-atomic: $x \mathcal C y \iff \exists a \exists b: a \ p^* \ x \land b \ p^* \ y \land a \mathcal C b$

Define: $ Cloud(X)=W \iff \operatorname {Unit}(W) \land \\ \forall l \ p^* \ W \exists a \ p \ X, \exists b \ p \ X: a \overset {l}- b \\ \land \forall a \ p^* \ X \, \forall b \ p^* \ X [a \neq b \implies \exists l \ p \ W: a \overset{l} - b] $

Define (Set-hood): $Set(X) \iff \\\exists Y: (Y \text { is fusion of Clouds } \lor Y=\varnothing) \land X=Cloud(Y) $

Now to make a system that can interpret $\sf ZFC$, we need to add size criterions:

we can coin an Axiom of Size:

If we add another size criterion that of Infinity, i.e, there is a Unit with infinite atoms. Then we get to interpret $\sf ZFC$.

C3-Atomic: $x \mathcal C y \iff \exists a \exists b: a \ p^* \ x \land b \ p^* \ y \land a \mathcal C b$

Define: $ Cloud(X)=W \iff \operatorname {Unit}(W) \land \\ \forall l \ p^* \ W \exists a \ p \ X \exists b \ p \ X: a \overset {l}- b \\ \land \forall a \ p^* \ X \, \forall b \ p^* \ X [a \neq b \implies \exists l \ p \ W: a \overset{l} - b] $

Define (Set-hood): $\begin{align}Set(X) \iff \exists Y: & (Y \text { is fusion of Clouds } \lor Y=\varnothing) \land \\ & X=Cloud(Y) \end{align} $

Now to make a system that can interpret $\sf ZFC$, we need to add size criterions. We can coin an Axiom of Size:

If we add another size criterion that of Infinity, i.e, there is a Unit composed of infinitely many atoms. Then we get to interpret $\sf ZFC$.

Define: $\operatorname {Unit}(x) \iff \\ \forall y \forall z (y \ p \ x \land z \ p \ x \land \operatorname {fusion}(y,z)=x \implies y \mathcal C z) \land \\ \forall k (\neg \mathcal O(k,x) \implies \neg k \mathcal C x)$

Define: $\begin{align} \operatorname {Unit}(x) \iff & \forall y \forall z (\operatorname {fusion}(y,z)=x \implies y \mathcal C z) \, \land \\& \forall k (\neg \mathcal O(k,x) \implies \neg k \mathcal C x)\end{align} $