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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?

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  • $\begingroup$ What is the difference between standard ordinal and non-standard ordinal? I do not see how to differentiate them. $\endgroup$
    – Hanul Jeon
    Commented Mar 15, 2021 at 9:27
  • $\begingroup$ A standard ordinal is a well founded transitive set of transitive sets. A non-standard ordinal is a transitive set of transitive sets that externally is not well founded but internally the theory thinks it's well founded, i.e. the theory from inside thinks it is an ordinal, but externally speaking it is not an ordinal because it is not well founded, i.e. externally there is a subset of it that is an infinite descending membership chain, but this subset is not captured internally by the theory, like the set of all elements of $\alpha$ that are moved by $j$ (where $\alpha$ is moved by j) $\endgroup$
    – Zuhair Al-Johar
    Commented Mar 15, 2021 at 9:37
  • $\begingroup$ You are working with a formal theory, but the distinction is made within a model. Can the difference between standard and non-standard ordinal be described over the first-order language of set theory? $\endgroup$
    – Hanul Jeon
    Commented Mar 15, 2021 at 16:57
  • $\begingroup$ from within the theory one cannot discriminate between standard and non-standard ordinals, the distinction is not made withIN a model, it is made outside it, from within the theory the phrase would be "for some ordinal $\alpha$ we have $j(\alpha) < \alpha$", I've written "non-standard" just to make things clearer from the outside. Ok, I'll change the phrasing to clarify this issue. $\endgroup$
    – Zuhair Al-Johar
    Commented Mar 15, 2021 at 17:59

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