User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20190803082138/@comment-11227630-20190803101521

My notion of \(\beta\text{-}\Pi_n[\eta]\)reflecting and \(\beta[\eta]\)stable (defined via reflecting) seems to match the initial part of gap ordinals.

Here are my guesses: In my notion, no ordinal α is 1[α+1]stable, so it stop here, but the concept of gap extends further.
 * 1[ω]stable : gap length 1
 * 1-П1[ω]reflecting : gap length 1 and also 1-П1reflecting
 * 2[ω]stable : gap length 1 and also (+2)-stable
 * 0-П1[ω+1]stable : "gap length 1" ordinals are stable to this nonprojectable ordinal
 * 0-П2[ω+1]stable : П2-(Gp)-reflecting
 * 1[ω+1]stable : gap length 1 and also (+1)-2-stable
 * 1[ω2]stable : gap length 2
 * α is 1[α]stable : α is gap length α