Let stratified $\sf Z$ have all axioms of $\sf Z$-$\sf Reg.$ with Infinity stipulated by existence of $\omega$, and with Separation restricted to use only stratified formulas, where those are defined after Quine's New Foundations. AC is not assumed.



 Define: $y = \mathcal P^{-1}(x) \iff x= \mathcal P(y)$

Define:  $x= \bigcap_{s \in S} s \iff  x=\{y \mid S \neq \varnothing \land \forall s \in S (y \in s)\}$

  Now, lets try coin what we mean by rank or stage of the inverted iterative hierarchy, which we'd label as *inverted rank*.

$X$ is an inverted rank, if and only if, there is an ordinal $\alpha$ and there is a bijective function $f$ whose domain is $\alpha$ that sends $0$ to $X$ and that sends each $\beta+1 \in \alpha$ to $ \mathcal P^{-1}(f(\beta))$, and sends each limit $\lambda \in \alpha$ to $ \bigcap_{\kappa < \lambda} f(\kappa)$.

We phrase that as: $X$ is an inverted rank of degree $\alpha$. 

For brevity "inverted rank" shall be denoted "i.rank".

Axiom I: $X \text { is an i.rank } \land Y \text{ is an i.rank} \to X \subseteq Y \lor Y \subseteq X$

Axiom II: $\forall x \exists Y: Y \text{ is an i.rank } \land x \in Y$

Axiom III: $\alpha \text { is an ordinal } \to \exists X: X \text{ is an i.rank of degree } \alpha$

Axiom VI: $\forall A \exists x \forall y \, (y \in x \leftrightarrow y \in A \land  y \text { is an i.rank } \}$

> Now, can this theory be consistent? I suspect the last axiom might cause problems, but even if so, can the theory without it be consistent? I mean this theory goes against the norms of the known cumulative hierarchy, where no descending power sequence like that is possible. So, does it stand a chance of being consistent whether with or without the last axiom.