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The folloing is an Edit of the previous question.

By stratified acyclic ZF, it is meant acylic ZF but with restricting Separation and Replacement to stratified instances of them and forbid the use of symbol $\mathcal R$ them.

Is Stratified acyclic ZF consistent with having an internal non-trivial automorphism $j$ from $V$ to $V$?

That is we add a new primitive one place function symbol $j$ to the language of ZF, and add the axioms of $j$ being an automorphism over the whole universe of discourse, that is:

Bijectivity: $\forall x \exists y : j(x)=y \\\forall y \exists x: j(x)=y$

Isomorphism: $\forall x \forall y :x \in y \iff j(x) \in j(y)$

Collapse: $\exists \alpha: V_\alpha \subsetneq V_{j(\alpha)}$

Where $``j"$ is allowed to appear in instances of the stratified separation and replacement.

If this is allowed what would be the consistency strength of such addition?

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  • $\begingroup$ Side question: is there anything known about the consistency strength of stratified $\mathsf{ZF}$? $\endgroup$
    – Hanul Jeon
    Commented Feb 10, 2021 at 23:29
  • $\begingroup$ @Hanul Jeon, what is known is that "stratified ZF + every set is the same size as a set of singletons", is equivalent to ZF. I'll try to find a source on that. However, I'm not sure of the strength of str ZF, perhaps it would be as strong as Zermelo or slightly stronger? $\endgroup$
    – Zuhair Al-Johar
    Commented Feb 11, 2021 at 8:03
  • $\begingroup$ Quick comment: $j$ cannot appear in the $\phi$s of the isomorphism axiom. $\endgroup$
    – Noah Schweber
    Commented Feb 11, 2021 at 15:07
  • $\begingroup$ You've omitted regularity now? So you can take ZFA with a non-empty set of atoms. Say, two atoms. Then the function that switches the atoms is a definable automorphism. And that's before we even stratified the axioms. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 11, 2021 at 18:15
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    $\begingroup$ That is not mentioned in the axiom. Also, since $j$ must map well-founded sets to well-founded sets, its restriction to the von Neumann universe would be an automorphism. Extensional and well-founded objects don't like automorphisms. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 11, 2021 at 18:43

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