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 $\kappa$?

  • 1
    $\begingroup$ 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
  • 3
    $\begingroup$ 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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.