The following question is about the possibility of having a world of sets obeying new foundations "**NF**" with their well founded sets obeying rules of **ZF**. It uses the revised version of Quines $``ML"$ ([Mathematical Logic (chapter 4)][1]) system, in order to define well foundedness in a faithful manner, and then adds axioms of size and infinity over the well founded sector. 

FORMAL EXPOSITION:

Language: First order logic with equality and membership with extra-logical axioms of: 

**Extensionality:** $\forall z (z \in x \leftrightarrow z \in y) \to x=y$

 **Classes:** if $\phi$ is a formula in which $x$ doesn't occur free, then $(\exists x \forall y (y \in x \leftrightarrow set(y) \land \phi))$

Where: $set(y) \iff \exists z(y \in z)$

Define: $x=V \iff \forall y \ (set(y) \to y \in x)$
 
**Stratification:** if $\phi(y,x_1,..,x_n)$ is a stratified formula in which $x$ doesn't occur free, and in which all quantifiers are bounded by $V$, and all free variables of it are among $y,x_1,..,x_n$ then:$$\forall x_1 \in V,...,\forall x_n \in V (\{y\in V| \phi\} \in V)$$

**Size:** $x,y\text{ are well founded} \land |x|=|y| \land x \in V \to y \in V$

Where: $\text{well founded} (x) \iff \\\not \exists d (x \cap d \neq \emptyset \land \forall m \in d \exists n \in d (n \in m))$

**Infinity:** $\omega \in V$

Where $\omega$ is the set of all finite Von Neumann ordinals.

>Question 1: Is this theory consistent relative to consistency of $NF$ and $ZF$?

>Question 2: if we weaken Extensionality to weak Extensionality of NFU, as to allow Ur-elements. Would that be consistent relative to NFU and ZFU.

 


  [1]: https://archive.org/details/QUINEMathematicalLogic/mode/2up