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

Oh. Then my analysis below is not correct. It should be like this:

C(Ω2+C(Ω2×ω,0),0) ~ start 1st 2nd-order gap length 1; 1st β|(Lβ/Lβ+1)∩P(ω)=∅; β|Lβ⊧Z2; β|Lβ⊧ZFC- C(Ω2+C(Ω2×α,0),0) ~ start 1st 2nd-order gap length α; 1st β|(Lβ/Lβ+α)∩P(ω)=∅ C(Ω2+C(Ω2×C(Ω2,0),0),0) ~ 1st β=(start 1st 2nd-order gap length β) = 1st β|(Lβ/Lβ+β)∩P(ω)=∅ C(Ω2+C(Ω2×C(Ω2,C(Ω2,0)),0),0) ~ 1st β=(start 1st 2nd-order gap length next admissible after β); 1st β|(Lβ/Lβ+)∩P(ω)=∅; β|Lβ⊧KP+∃P(ω) C(Ω2+C(Ω2×C(Ω2,C(Ω2,C(Ω2,0)))),0),0) ~ 1st β=(start 1st 2nd-order gap length two next admissible after β); 1st β|(Lβ/Lβ++)∩P(ω)=∅ C(Ω2+C(Ω2×C(Ω2+1,0),0),0) ~ 1st limit of β=(start 1st 2nd-order gap length n<ω admissible after β); C(Ω2+C(Ω2×C(Ω2+C(Ω2×2+1,0),0),0),0) ~ 1st limit of β=(start 1st 2nd-order gap length (n<ω)-ple stable after β); C(Ω2+C(Ω2×C(Ω2+C(Ω2×ω,0),0),0),0) ~ start 1st 2nd-order gap length β=(1st 2nd-order-gap with length "next 2nd-order gap ordinal after β"); C(Ω2+C(Ω2×C(Ω2×2+1,0),0),0) ~ 1st limit of β<α|(Lβ/Lα)∩P(ω)=∅; nonprojectable 2nd-order gap; 0-П1-reflecting on class Gap-ordinals; Σ1-admissible on class Gap-ordinals C(Ω2+C(Ω2×C(Ω2×3+1,0),0),0) ~ 0-[2]П1-reflecting on class Gap-ordinals; Σ2-admissible on class Gap-ordinals C(Ω2+C(Ω2×C(Ω2×ω,0),0),0) ~ 2nd-order gap length β on class Gap-ordinals; 2nd-order 2-gap length β C(Ω2+C(Ω2×C(Ω2×C(Ω2×2+1,0),0),0),0) ~ 0-П1-reflecting on class 2-Gap-ordinals; Σ1-admissible on class 2-Gap-ordinals C(Ω2+C(Ω2×C(Ω2×C(Ω2×3+1,0),0),0),0) ~ 0-[2]П1-reflecting on class 2-Gap-ordinals; Σ2-admissible on class 2-Gap-ordinals C(Ω2+C(Ω22,0),0) ~ start 1st 3d-order gap length 1; 2nd-order ω-gap length β β|(Lβ/Lβ+1)∩P(P(ω))=∅; β|Lβ⊧Z3; β|Lβ⊧ZFC-+∃P(ω) C(Ω2+C(Ω22,C(Ω22,0)),0) ~ start 1st 4th-order gap length 1; 1st β|(Lβ/Lβ+1)∩P(P(P(ω)))=∅; β|Lβ⊧Z4; β|Lβ⊧ZFC-+∃P(P(ω)) C(Ω2+C(Ω22+1,0),0) ~ β|Lβ⊧Zn; β|Lβ⊧ZFC-+∀n∃ωn+V=L; β|Lβ⊧ZFC-+∃1st П1-reflecting on class P(n)-ordinals C(Ω2+C(Ω22,C(Ω22+1,0)),0) ~ β|Lβ⊧ZFC-+∃ωω+V=L C(Ω2+C(Ω22+C(Ω2×C(Ω22,0),0),0),0) ~ β|Lβ⊧ZFC-+∃ωω 1 +V=L C(Ω2+C(Ω22+C(Ω2×C(Ω22+1,0),0),0),0) ~ β|Lβ⊧ZFC-+∀n∃ωω n +V=L C(Ω2+C(Ω22+Ω2,0),0) ~ β|Lβ⊧ZFC-+∃Beth fixed point C(Ω2+C(Ω2,C(Ω22+Ω2,0),0),0) ~ β|Lβ⊧ZFC-+∃power-admissible; β|Lβ⊧ZFC-+∃1st П2-reflecting on class P(n)-ordinals

This assumption is consistent with Taranovsky’s analysis (6.2). But then in this analysis he uses Σ2-extendible(γ|Vγ≺2Vγ,undefined where k - inaccessible cardinal). C(Ω2+C(Ω2×2,C(Ω22+Ω2,0),0),0) ~ β|Lβ⊧ZFC-+∃Σ2-extendible(γ|Vγ≺2Vγ,undefined where k - inaccessible cardinal) C(Ω2+C(Ω2×3,C(Ω22+Ω2,0),0),0) ~ β|Lβ⊧ZFC-+∃Σ3-extendible(β|Vγ≺3Vγ,undefined where k - inaccessible cardinal) C(Ω2+C(Ω2×ω,C(Ω22+Ω2,0),0),0) ~ β|Lβ⊧ZFC; β|Lβ⊧ZFC-+∃γ|(Vγ/Vγ+1)∩Vk=∅(where k - inaccessible cardinal); β|Lβ⊧ZFC-+∃gap of cardinality length 1 exists; β|Lβ⊧ZFC-+∃1st worldly cardinal

And I don’t understand how to go to this ordinal. Σ2-extendible is power-recursive inaccessible = П2-reflecting that is П1-reflecting on П2-reflecting on class P(n)-ordinals? or Σ2-extendible is like nonprojectable on class P(n)-ordinals? or Σ2-extendible is like (0-[2]П1-reflecting) Σ2-admissible on class P(n)-ordinals? Or even bigger?