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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 $\kappa$?

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    $\begingroup$ If you write 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. $\endgroup$ Feb 9 at 23:39
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    $\begingroup$ But it should be \cdots..., no? $\endgroup$ Feb 10 at 0:36
  • $\begingroup$ @MichaelHardy, Thanks a lot for these corrections. $\endgroup$ Feb 10 at 6:34

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