User blog comment:Hyp cos/Attempt of OCF up to Stability/@comment-31580368-20181029042115/@comment-11227630-20181029083725

Unfortunately, this notation can't express sup{(x+n)-stable|n<λx.x-stable}. However, the (+S(1,0))-stable ordinal (the least one is S(1,0) itself) will work as the actual sup{(x+n)-stable|n<λx.x-stable}; the (+S(1,0)+1)-stable ordinal (the least one is S(S(1,0)+1)) will work as the actual (+S(1,0))-stable ordinal, and so on. So it doesn't affect the final strength.