# Can we have external automorphisms over intersectional models?

Is the following inconsistent:

By "intersectional" set I mean a set having the intersectional set of every nonempty subset of it, being an element of it.

$$\forall S \subset M: S\neq \varnothing \to \bigcap S \in M$$

Where: $$\bigcap S = \{x \mid \forall s \in S \, (x \in s) \}$$

Sets $$(\varnothing, \{\varnothing\}, V_\omega, H_x, \operatorname {Fin}(\omega.2 \setminus \omega))$$ are examples; where $$H_x$$ is the set of all sets hereditarily strictly subnumerous to $$x$$; and $$\operatorname {Fin}(x)$$ is the set of all finite subsets of $$x$$.

Let the ambient theory of models be $$\sf ZF$$-$$\sf Reg.$$. Can we have a model $$M$$ of say Mac Lane set theory (with or without Regularity, and without Choice) that admits a non-trivial external automorphism and at the same time have $$M$$ be an intersectional set?

• Do you intend that the membership relation of $M$ is the ambient membership relation $\in$? Commented May 10 at 11:25
• Also, I guess you intend that the automorphism is nontrivial. Commented May 10 at 11:25
• @JoelDavidHamkins, Yes! For both, but the ambient theory of models is ZF-Reg. Also for Mac Lane set theory I take the version that doesn't have Regularity among its axioms. I've edited. Thanks! Commented May 10 at 11:29
• Ah, sorry, I missed the Reg part. Why use such a weird ambient theory? With reg, no standard model has any nontrivial automorphism at all, and intersectionality is irrelevant. Commented May 10 at 11:49
• @JoelDavidHamkins, I know! That's why you cannot have the ambient theory be ZF, since I need $\in$ to be the one of the ambient theory. I also made the typo of writing it first as $\sf ZF$, but it should be $\sf ZF-Reg$. Commented May 10 at 12:22

One can easily make a model of ZF-Reg with numerous Quine atoms. Simply begin with a model of ZFCU, with numerous urelements, and then turn the urlements into Quine atoms, which are singleton sets $$a=\{a\}$$, and the result is a model of ZF-Reg. The atoms can be permuted, and these permutations extend to automorphisms of the whole universe.
For example, consider $$M=V_{\omega_1}[A]$$ in the original universe, with a set $$A$$ of urelements. In the new model, these turn into Quine atoms, and this model $$M$$ is supertransitive, hence intersectional, and a model of MacLane set theory (without Reg) and much more. But permutations of $$A$$ extend to automorphisms of $$M$$.
• One important issue, can one of these automorphisms be rank shifting? That is, it can shift some non-standard ordinal $\alpha$ say inwardly? That is, we have $j(\alpha)< \alpha$, for an automorphism $j$? Commented May 10 at 13:21