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

We add a new primitive unary function symbol $U$ to the language of set theory.

The idea is to have a hierarchy $$\cdots \subset U_{-2} \subset U_{-1} \subset U_0 \subset U_1 \subset U_2 \subset \cdots$$, where each stage (rank) $U_{i+1}$ is the powerset of its predecessor $U_{i}$

I call this "inverted hierarchy" because ranks can have an infinite descent.

Axiom I: $\forall i \in \mathbb Z: U_i \subset U_{i+1} \land U_{i+1} = \mathcal P (U_i) $

We call each $U_i$ an ultra-rank, denoted "$u$.rank"

Axiom II: $\forall x \exists i \in \mathbb Z: x \in U_i$

Axiom III: $\forall i \in \mathbb Z \exists x: x=U_i$

Axiom VI: $ \forall i \in\mathbb Z: \forall \alpha \, ( V_\alpha \in U_i)$

Where $V_\alpha$ is defined in the usual manner, i.e. as the $\alpha^{th}$ iterative power of $\varnothing$.

[NOTE]: $\mathbb Z$ is internal. The $+$ operator on $\mathbb Z$ is defined so that $i+1$ is one type higher than $i$ in stratification. For example we take $\mathbb Z$ to be the set of all acyclic infinitely iterative singletons $\{\{\{\cdots \space \cdots\}\}\}$, fix one of its elements to be $0$ and define $i+1 = \{i\}$. Needed axioms to support such internal grasp of $\mathbb Z$ to be added. $U$ can be used in Separation. The arguments of $U$ receive equal type indices during stratification, so $U_i$ and $i$ receive the same type.

If the above is consistent can we have an external automorphism that shifts $u$.ranks one-step downwardly? That is, there is an external automorphism $j$ such that for some $i \in \mathbb Z$, we have: $U_{j(\alpha) + 1} = U_\alpha$, for all $\alpha \in \mathbb Z, \alpha < i $.

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    $\begingroup$ You seem to be interested in models of set theory with automorphisms that change rank. This can happen even in models of ZFC (which extends your theory). Take any computably saturated countable model---it has many automorphisms that change rank on the ordinals. The same is true for any saturated model. (These automorphisms never change the rank by exactly one, however, since ordinals follow the even/odd pattern.) $\endgroup$
    – Joel David Hamkins
    Commented May 12 at 13:57
  • $\begingroup$ Can you let us know whether $U$ is allowed to appear in formulas in the separation scheme? And you use $\mathbb{Z}$ as indices, but is this meant to be the internal version of $\mathbb{Z}$ or the metatheoretic $\mathbb{Z}$? $\endgroup$
    – Joel David Hamkins
    Commented May 12 at 14:40
  • $\begingroup$ @JoelDavidHamkins, $\mathbb Z$ is internal. The $+$ operator on $\mathbb Z$ is defined so that $i+1$ is one type higher than $i$ in stratification. $U$ can be used in Separation. The arguments of $U$ receive equal type indices during stratification, so $U_i$ and $i$ receive the same type. For example we take $\mathbb Z$ to be the set of all acyclic infinitely iterative singletons $\{\{\{\cdots \space \cdots\}\}\}$, fix one of its elements to be $0$ and define $i+1 = \{i\}$. Of course we need to add axioms to support such internal grasp of $\mathbb Z$. $\endgroup$
    – Zuhair Al-Johar
    Commented May 12 at 16:01
  • $\begingroup$ @JoelDavidHamkins, one of the aims from developing such a hierarchy is to remove the obstacles to one-type shift automorphisms. I hope the even/odd problem can be bypassed. $\endgroup$
    – Zuhair Al-Johar
    Commented May 12 at 18:00
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    $\begingroup$ Well, I wish you good luck. My view is that when one finds that one's foundational theory allows for bizarre undesirable phenomenon, it is a reason to adopt a better theory. $\endgroup$
    – Joel David Hamkins
    Commented May 12 at 18:26

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