User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20191108142819/@comment-31580368-20191109095511

class "Gap-ordinals" means "Gap-ordinals bellow" or class {β<α|(Lβ/Lα)∩P(ω)=∅} or Gp

1st limit of β<α|(Lβ/Lα)∩P(ω)=∅ is 0-П1-reflecting on chain β<α|(Lβ/Lα)∩P(ω)=∅ or Σ1-admissible on chain β<α|(Lβ/Lα)∩P(ω)=∅

If suppose that П2-(Gp)-reflecting or 0-П2-reflecting on chain β<α|(Lβ/Lα)∩P(ω)=∅ is 3rd-order Δ2-gap

Then we can continue the transformation further 0-Пω-reflecting on chain β<α|(Lβ/Lα)∩P(ω)=∅ 1-П1-reflecting on chain β<α|(Lβ/Lα)∩P(ω)=∅ 0-[2]П1-reflecting on chain β<α|(Lβ/Lα)∩P(ω)=∅ or Σ2-admissible on chain β<α|(Lβ/Lα)∩P(ω)=∅ 0-[3]П1-reflecting on chain β<α|(Lβ/Lα)∩P(ω)=∅ or Σ3-admissible on chain β<α|(Lβ/Lα)∩P(ω)=∅ 0-[ω]П1-reflecting on chain β<α|(Lβ/Lα)∩P(ω)=∅ or Σω-admissible on chain β<α|(Lβ/Lα)∩P(ω)=∅ is equal 2nd-order gap length 1 on chain β<α|(Lβ/Lα)∩P(ω)=∅ or 2nd-order 2-gap length 1

After: class "2-Gap-ordinals" or "2Gp" means class {β<α|(Lβ/Lα)∩P(ω)=∅} on chain β<α|(Lβ/Lα)∩P(ω)=∅ П2-(2Gp)-reflecting is 3rd-order Δ3-gap in finally 3rd-order Δω-gap is 3rd-order gap length 1 or β|Lβ⊧Z3

I made this assumption in accordance with the above chain of diagonalizations of TON expressions. Taranovsky in his analysis concludes that Z3should already be C(Ω22,0).