is below that of Π-indescribable, but for m>1 it is consistent with ZFC that the least Σ-indescribable exists and is above the least Π-indescribable cardinal (this is proved from consistency of ZFC with Π-indescribable cardinal and a Σ-indescribable cardinal above it). Measurable cardinals are Π-indescribable, but the smallest measurable cardinal is not Σ-indescribable. However, assuming choice, there are many totally indescribable cardinals below any measurable cardinal. Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for Π and Σ indescribability. References Large cardinals