# Is it consistent to have these kinds of acyclic hereditarily size sets?

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 $$\kappa$$?

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