Taken from Math Stack Exchange.



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> 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 $\textrm{Reg}$.
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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?



Equivalences of $\mathcal{F}$-Mahloness
The paper ["Small Definably-large Cardinals" by Roger Bosch](https://doi.org/10.1007/3-7643-7692-9_3) 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.


