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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?

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    $\begingroup$ By the way, questions like this might fit better at math.stackexchange in the future. $\endgroup$
    – Noah Schweber
    Commented Oct 20, 2017 at 2:18

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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.

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