Why is it sometimes called 1-consistency? $\Sigma_1$-soundness of a theory (meaning that every theorem it proves that is $\Sigma_1$ in the arithmetical hierarchy is true in the natural numbers) is sometimes called 1-consistency, which suggests that it is in a hierarchy of notions called $n$-consistency, with $n$ ranging over the natural numbers. Is there such a hierarchy, and if there is, is $\omega$-consistency some kind of limit of that hierarchy?

Daniel Isaacson's lectures on the Gödel's Incompleteness Theorems (which Mauro Allegranza linked in a comment) mention that $n$-consistency is defined as $\omega$-consistency with respect to $\Sigma_n$-sentences and that this terminology, or at least 1-consistency, was introduced by Kreisel.
