Working in $\sf ZFC-Reg. {}+ Acyclicity$. Where :

**Acyclicity:** $\neg \exists x_1, \cdots, \exists x_n: x_1 \in x_2 \in\cdots\in x_n \in x_1$

We add the following kind of weird non-well founded sets.

$\forall \kappa \, \exists y: \forall z \in \operatorname {Tc}(\{y\}) (|z| = \kappa)$

where $\kappa$ is a Scott cardinal. $\operatorname {Tc}$ stands for transitive closure defined as the minimal transitive superset. $|\cdot|$ stands for cardinality defined after Scott's.

Define: $\mathcal H_\kappa(y) \iff \forall z \in \operatorname {Tc}(\{y\}) (|z| = \kappa)$

Can we also add the following rather strange assertion?

$$\forall \kappa < \omega \, \forall x \, \forall y: \\ \mathcal H_\kappa (x) \land \mathcal H_\kappa (y) \land x \neq y \to x \in \operatorname{Tc}(y) \lor y \in \operatorname{Tc}(x) $$

Is this consistent?

Can we have that rule for any $\kappa$?

`x_2\in...\in x_n`

in LaTeX you'll see something like ${x_2\in}\text{...}{\in x_n},$ whereas MathJax often corrects that so that you see $x_2\in\ldots\in x_n$ instead. However, that doesn't seem to work between two instances of`\in`

(i.e. $\in$) the way it does with plus signs and most other things, so you see instead $x_2\in...\in x_n,$ with the dots closer to the first $\in$ than to the second, and closer to both than it should be. I put in`\ldots`

. $\endgroup$