User blog comment:Scorcher007/Hierarchy of ordinals Z 2. Help me to understand/@comment-25601061-20180825031717

There doesn't exist an α that's α-indescribable, but you can have things like the first (first (first (...)-indescribable)-indescribable)-indescribable. As for x-le stables, if they're the stables with ≺1 chains of at least length x, then ω-le stables are the same as nonprojectibles, (ω+1)-le stables get us to the PTO of П12-CA+BI, and (ε0)-le stables get us to the PTO of ∆13-CA (source)