Working in some suitable extension of ZF, can we have a sequence of external [not appear in instances of separation and replacement] automorphisms $(j_n)_{n \in \omega}$ over the universe that move ranks dowardly, i.e. for some non-standard ordinal $\alpha$ we have $j(\alpha) < \alpha$ and $V_{j(\alpha)} \subsetneq V_\alpha$, in such a manner that there exists a limit stage $V_\alpha$ for $\alpha = \beta + \gamma $ for some countable limit ordinal $\gamma$, and we have: $$V_{j_n(\alpha)+1} \subsetneq V_{j_{n+1}(\alpha) +1} \\ \bigcup^{n \in \omega} V_{j_n(\alpha)+1} = V_\alpha $$

A related question is if the above is possible, can that be done in an extension of ZF in which Global choice hold?

Working in some suitable extension of ZF, can we have a sequence of external [not appear in instances of separation and replacement] automorphisms $(j_n)_{n \in \omega}$ over the universe that move ranks dowardly, i.e. for some ordinal $\alpha$ [externally non-standard] we have $j(\alpha) < \alpha$ and $V_{j(\alpha)} \subsetneq V_\alpha$, in such a manner that there exists a limit stage $V_\alpha$ for $\alpha = \beta + \gamma $ for some countable limit ordinal $\gamma$, and we have: $$V_{j_n(\alpha)+1} \subsetneq V_{j_{n+1}(\alpha) +1} \\ \bigcup^{n \in \omega} V_{j_n(\alpha)+1} = V_\alpha $$

A related question is if the above is possible, can that be done in an extension of ZF in which Global choice hold?

Working in some suitable extension of ZF, can we have a sequence of external [not appear in instances of separation and replacement] automorphisms $(j_n)_{n \in \omega}$ over the universe that move ranks dowardly, i.e. for some non-standard ordinal $\alpha$ we have $j(\alpha) < \alpha$ and $V_{j(\alpha)} \subsetneq V_\alpha$, in such a manner that there exists a limit stage $V_\alpha$ for $\alpha = \beta + \omega $ , and we have: $$V_{j_n(\alpha)+1} \subsetneq V_{j_{n+1}(\alpha) +1} \\ \bigcup^{n \in \omega} V_{j_n(\alpha)+1} = V_\alpha $$

A related question is if the above is possible, can that be done in an extension of ZF in which Global choice hold?

Working in some suitable extension of ZF, can we have a sequence of external [not appear in instances of separation and replacement] automorphisms $(j_n)_{n \in \omega}$ over the universe that move ranks dowardly, i.e. for some non-standard ordinal $\alpha$ we have $j(\alpha) < \alpha$ and $V_{j(\alpha)} \subsetneq V_\alpha$, in such a manner that there exists a limit stage $V_\alpha$ for $\alpha = \beta + \gamma $ for some countable limit ordinal $\gamma$, and we have: $$V_{j_n(\alpha)+1} \subsetneq V_{j_{n+1}(\alpha) +1} \\ \bigcup^{n \in \omega} V_{j_n(\alpha)+1} = V_\alpha $$

A related question is if the above is possible, can that be done in an extension of ZF in which Global choice hold?