User blog comment:Clarrity/List of ordinals (up to omega^omega)/@comment-30118230-20180225131945/@comment-30754445-20180225200616

Unfortunately, it is theoretically impossible to give a general form for all ordinals up to ω₁ck because ω₁ck isn't recursive.

For any given limit that is strictly below ω₁ck, it can (theoretically) be done. But for ω₁ck itself - it can't.