Equivalences of $\mathcal{F}$-Mahloness

Question

Taken from Math Stack Exchange.

Let $\mathcal{F}$ be a set of $\mathcal{L}_\in$-formulae, $\kappa$ be a cardinal and $A \subset \textrm{Ord}$. Then, $\kappa$ is called $\mathcal{F}$-Mahlo if $A \cap \kappa$ intersects every club definable in $H_\kappa$ by a formula $\varphi \in \mathcal{F}$. $\kappa$ is $\mathcal{F}$-Mahlo if it is $\mathcal{F}$-Mahlo onto $\textrm{Reg}$.

This has some interesting properties. For example, if we let $\Pi$ denote the standard Levy hierarchy, then every $\Pi_1$-Mahlo cardinal is a weakly inaccessible limit of weakly inaccessible cardinals, i.e. weakly 2-inaccessible. Now, a well-known result is that $\kappa$ is $\Pi^1_n$-indescribable iff it is $\Sigma^1_{n+1}$-indescribable (a similar thing applies to reflecting ordinals). Does this apply to Mahloness? In other words, is $\kappa$ $\Pi_n$-Mahlo iff it is $\Sigma_{n+1}$-Mahlo? Also, does any kind of similar equivalence apply to $\Delta_n$-Mahloness?

Answer 1

The paper "Small Definably-large Cardinals" by Roger Bosch proves that an inaccessible cardinal is $\Sigma_{n+1}$-Mahlo if and only if it is $\Pi_n$-Mahlo except for $n=1$ (I'm referring to the boldface hierarchy, not the lightface hierarchy which is less relevant to your question) and that an inaccessible cardinal is $\Sigma_2$-Mahlo if and only if it is $\Delta_2$-Mahlo. Whether $\Pi_1$-Mahlo cardinals are always $\Delta_2$-Mahlo (and thus $\Sigma_2$-Mahlo) an open problem as far as I know.