I am interested in the existence of the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ for any given set $x$, in the context of NFU (New Foundations with Urelements). It seems to me that the comprehension axiom in NFU does not allow to prove that $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists for any set $x$. On the other hand, it seems to me that the model of NFU described in the SEP article Alternative Axiomatic Set Theories satisfies the property that for every $x$, $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists.

Let me informally describe my reasons for believing that in that model, $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists for every $x$. The model described in the SEP article is defined with the help of a non-standard model of ZFC with an automorphism $j$ that moves rank $\alpha$ to a lower rank. In that case, the $V_\alpha$ of that model of ZFC is a model of NFU, if $\in_{NFU}$ is suitably defined as in the linked article. Under those conditions, the interpretation of $\mathcal{P}(x)$ in the model will be an element of $V_{j(\alpha)+1}$. But then the interpretation of $\mathcal{P}^n(x)$ in the model will be an element of $V_{j(\alpha)+1}$ for any $n \in \mathbb{N}$, so the interpretation of $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ in the model will also be an element of $V_{j(\alpha)+1}$, so $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists according to the model.

I have four interdependent questions:

- Is it correct that the statement "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" is not provable in NFU?
- Is my above sketch of the claim that the statement "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" could be added to NFU without losing consistency correct?
- If the answer to the second question is "no", are there other known ways to prove that the statement "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" could be added to NFU without losing consistency?
- If the answer to the fourth question is "no", is it known that adding "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" to NFU leads to an inconsistency?