# Can there be a proper class of $\alpha$-inaccessibles for any ordinal $\alpha$?

A cardinal $$\kappa$$ is $$\alpha$$-inaccessible if $$\kappa$$ is inaccessible and such that for any ordinal $$\beta < \alpha$$, the set of $$\beta$$-inaccessibles less than $$\kappa$$ has cardinality $$\kappa$$.

For example, a cardinal is 1-inaccessible if it is an inaccessible limit of (0)-inaccessibles. A domain $$V_{\kappa_1}$$ for $${\kappa_1}$$ a 1-inaccessible would have a proper class of 0- inaccessibles. Likewise, a domain $$V_{\kappa_2}$$ for $$\kappa_2$$ a 2-inaccessible would contain a proper class of 1-inaccessibles.

However, I think there must be some ordinals $$\gamma$$ for which there cannot be a proper class of $$\gamma$$-inaccessibles, but I cannot make this intuition precise enough to prove it. If my intuition is correct, for which ordinals does it fail?

• There is no such $\gamma$. That is, we expect significantly stronger large cardinal assumptions to be consistent. Sep 29, 2018 at 17:24

If $$\kappa$$ is Mahlo, then $$\kappa$$ is $$\kappa$$-inaccessible. So if there is a Mahlo cardinal $$\kappa$$, we get that $$V_\kappa\models$$ "There is a proper class of $$\alpha$$-inaccessibles for every ordinal $$\alpha$$." Since the existence of Mahlo cardinals is not currently known to be inconsistent with ZFC - to put it mildly (Mahlo cardinals are incredibly low down in the large cardinal hierarchy) - the answer to your question is (currently) no.
In fact, even the existence of a proper class of Mahlos isn't much to write home about: it holds in $$V_\lambda$$ whenever $$\lambda$$ is weakly compact.
• If $\kappa$ is Mahlo, then it is a limit of $\lambda$-inaccessibles for all $\lambda<\kappa$. Sep 29, 2018 at 17:41