User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20180930023421

Information on the highest degrees of stability is very small. I found them only at M.Rathjen (Proposition 5.3 - 5.6, p.19) and at T.Arai (9 - 9.0, p.13-14). I'm using my notation to write ordinals, I can assume that it works like this: Ω = S[a] - 1st admissible after ω, Ω(1) = S[a](1) - 2nd admissible after ω, e.t.c. Ω(α) = S[a](α) - 1st fixed point of admissibles. I = S[a,1] - 1st recursively inaccessible or Δ11-set of term on degrees of fixed point of admissibles. I(1) = S[a,1](2) - 2nd recursively inaccessible. M(1) = S[a,2] - 1st recursively Mahlo or Δ11-set of term on degrees of recursively hyper-inaccessible. S[a,3] - 1st П3-reflection or Δ11-set of term on degrees of recursively hyper-Mahlo. And so on. S[a,ω] = S[a+1] - 1st (+1)-stable. S[a+1,1] - 1st 1-П1-reflection or Δ11-set of term on degrees of recursively hyper-(+1)-stable. S[a+2] - 1st (+2)-stable.

S[Ω(a+1)] = S[S[a(2)](a+1)] - (+)-stable. S[Ω(a+2)] = S[S[a(2)](a+2)] - (++)-stable. S[I(a+1)] = S[S[a(2),1](a+1)] - inaccessibly stable. S[M(a+1)] = S[S[a(2),2](a+1)] - Mahlo stable. S[S[a(2)+1](a+1)] = S[a(2)+1] - double (+1)-stable. S[S[S[a(3)+1]a(2)+1]a+1] = S[a(3)+1] - triple (+1)-stable. S[a(ω)+1] - <ω-ple (+1)-stable (П12-CA0). S[a(ω+1)+1] - ω-ple (+1)-stable (П12-CA+BI, nonprojectable). S[a(ε0)+1] - <ε0-ple (+1)-stable (Δ12-CA).

After accordind T.Arai Δ12-CA+BI ordinal is ordinal x such x is П2(St)-reflecting, where St denotes the set of stable below x. I believe in terms of a-ple stability it means Δ11-set of term on degrees of fixed point of S[a(α)+1]. Like 1st recursively inaccessible on fixed point of admissibles. I can write it this way S[S2[a,1]] = S2[a,1]. Then S[S2[a,1]+1] means next stability after S[S2[a,1]]; S[a(S2[a,1]+1)] means next n-ple stability after S[S2[a,1]];  S[S2[a,1](1)] = S2[a,1](1) means next inaccessibility of fixed point a-ple stability; e.t.c. S2[a,2], S2[a,3], ..., S2[a,ω] = S2[a+1] - 1st 2-stable. And so on... S2[a(ω)+1] - П13-CA0, S3[a(ω)+1] - П13-CA0, ..., ω-stable - Z2.