Least non-arithmetical ordinal As I understand, there exists the least ordinal $\alpha$ such that there is no well-ordering of $\mathbb{N}$ which is both order isomorphic to $\alpha$ and is an arithmetical set. Is there a conventional name for that ordinal? Is every ordinal above $\alpha$ non-arithmetical as well?

As Joel David Hamkins pointed out to me in another question on MO, the set of arithmetical ordinals is exactly the set of recursive ordinals $\omega^{\mathrm{CK}}_1$. See <