# Equivalences of $\mathcal{F}$-Mahloness

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?

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.
• I mean that I'm referring to boldface $\Pi_n$- and $\Sigma_n$-Mahlo cardinals, as opposed to the lightface $\Pi_n$- and $\Sigma_n$-Mahlo cardinals, which are defined using only clubs that are definable without parameters, since your definitions is the same as Bosch's boldface $\Pi_n$- and $\Sigma_n$-Mahlo cardinals. Apr 25, 2022 at 14:10