It is quite well-known that $\Pi\_2^0$-Indescribability is the same as Strong Inaccessibility and $\Pi\_n^0$-Indescribability for every $n>2$.


It is also quite simple to show that $\Pi\_0^0$-Indescribability is equivalent to nonemptiness (i.e. $\kappa>0$).


However, this gives some relations for almost every $\Pi\_n^0$ for finite $n$, but not completely. Specifically, the only Large Cardinal Class missing is $\Pi\_1^0$-Indescribable cardinals. Are these cardinals equivalent to some large cardinal property less than Inaccessibility? Are these cardinals trivial, and can ZFC prove they exist?



What is known about $\Pi_1^0$-Indescribable cardinals?
A $\Pi\_1$ statement $\varphi$ is absolute downwards between transitive structures: if $(V\_\alpha; \in, A)\models\varphi$ and $\beta<\alpha$, then $(V\_\beta;\in, A\cap V\_\beta)\models\varphi$. So $\Pi^0\_1$-indescribability, like $\Pi^0\_0$-indescribability, is trivial.


