Let $ZFC^{-}$ be the theory of $ZFC$ minus the axiom of foundation and define the proper classes $G$ and $V$ as follows:

$G:=$ The proper class of all sets.

$V:=$ The proper class of Von neumann cumulative heirachy.

And let the statments $G=V$ and $|G|=|V|$ be:

$G=V~:~~\forall x~\exists y~(ord(y)~ \wedge "x\in V_{y}")$

$|G|=|V|~:~~\exists F:G \longrightarrow V~~$ a one to one function.

Using the axiom of foundation it is clear that we have $G=V$ and $|G|=|V|$ means that G is as "small" as Von neumann's cumulative heirachy.

Now the question is: "How big is the proper class of all sets in the absence of the axiom of foundation? "

In the other words which one of the following statements are true?

(1) $Con(ZFC) \longrightarrow Con(ZFC^{-} + G\neq V)$

(2) $Con (ZFC) \longrightarrow Con(ZFC^{-} + |G|\neq|V|)$

Let $\operatorname{ZFC}^{-}$ be the theory of $\operatorname{ZFC}$ minus the axiom of foundation and define the proper classes $G$ and $V$ as follows:

$G:=$ The proper class of all sets.

$V:=$ The proper class of Von neumann cumulative heirachy.

And let the statements $G=V$ and $|G|=|V|$ be:

$G=V~:~~\forall x~\exists y~(\operatorname{ord}(y)~ \wedge "x\in V_{y}")$

$|G|=|V|~:~~\exists F\colon G \longrightarrow V~~$ a one to one function.

Using the axiom of foundation it is clear that we have $G=V$ and $|G|=|V|$ means that $G$ is as "small" as Von Neumann's cumulative hierarchy.

Now the question is: "How big is the proper class of all sets in the absence of the axiom of foundation? "

In the other words which one of the following statements are true?

(1) $\operatorname{Con}(\operatorname{ZFC}) \longrightarrow \operatorname{Con}(\operatorname{ZFC}^{-} + G\neq V)$

(2) $\operatorname{Con}(\operatorname{ZFC}) \longrightarrow \operatorname{Con}(\operatorname{ZFC}^{-} + |G|\neq|V|)$