User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20191006023842

My attempt for TON and LCO comparisons from Z2 to ZFC and further

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×ω×2,0),0) - start 1st 2nd-order gap length 2; 1st β|(Lβ/Lβ+2)∩P(ω)=∅ 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); β|Lβ⊧П21-CA0 C(Ω2+C(Ω2×C(Ω2×2+1,0),0),0) -1st limit of β=(start 1st 2nd-order gap length (n<ω)-ple stable after β); β|Lβ⊧П22-CA0 C(Ω2+C(Ω2×C(Ω2×ω,0),0),0) - start 1st 3d-order gap length 1; 1st β|(Lβ/Lβ+1)∩P(P(ω))=∅; β|Lβ⊧Z3; β|Lβ⊧ZFC-+∃P(ω) C(Ω2+C(Ω2×C(Ω2×C(Ω2×ω,0),0),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(Ω2×C(Ω22,0),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(Ω2×C(Ω22+1,0),0),0) - β|Lβ⊧ZFC-+∃ωω+V=L C(Ω2+C(Ω2×C(Ω22+C(Ω2×ω,0),0),0),0) - β|Lβ⊧ZFC-+∃ωω 1 +V=L C(Ω2+C(Ω2×C(Ω22+C(Ω2×C(Ω22,0),0),0),0),0) - β|Lβ⊧ZFC-+∀n∃ωω n +V=L C(Ω2+C(Ω2×C(Ω22+C(Ω22+Ω2,0),0),0),0) - β|Lβ⊧ZFC-+∃Beth fixed point C(Ω2+C(Ω2×C(Ω22+C(Ω2,C(Ω22+Ω2,0),0),0),0),0) - β|Lβ⊧ZFC-+∃power-admissible; β|Lβ⊧ZFC-+∃1st П2-reflecting on class P(n)-ordinals C(Ω2+C(Ω2×C(Ω22+C(Ω2×2,C(Ω22+Ω2,0),0),0),0),0) - β|Lβ⊧ZFC-+∃Σ2-extendible(γ|Vγ≺2Vγ,undefined where k - inaccessible cardinal) C(Ω2+C(Ω2×C(Ω22+C(Ω2×3,C(Ω22+Ω2,0),0),0),0),0) - β|Lβ⊧ZFC-+∃Σ3-extendible(β|Vγ≺3Vγ,undefined where k - inaccessible cardinal) C(Ω2+C(Ω2×C(Ω22+C(Ω2×ω,C(Ω22+Ω2,0),0),0),0),0) - β|Lβ⊧ZFC; β|Lβ⊧ZFC-+∃γ|(Vγ/Vγ+1)∩Vk=∅(where k - inaccessible cardinal); β|Lβ⊧ZFC-+∃least cardinal β that is not definable in ZFC; β|Lβ⊧ZFC-+∃gap of cardinality length 1 exists; β|Lβ⊧ZFC-+∃1st worldly cardinal C(Ω2+C(Ω2×C(Ω22+C(Ω2×ω×2,C(Ω22+Ω2,0),0),0),0),0) - β|Lβ⊧ZFC-+∃γ|(Vγ/Vγ+2)∩Vk=∅(where k - inaccessible cardinal); β|Lβ⊧ZFC-+∃gap of cardinality length 2 exists; β|Lβ⊧ZFC-+∃2nd worldly cardinal C(Ω2+C(Ω2×C(Ω22+C(Ω2×C(Ω2×ω,C(Ω22+Ω2,0),0),C(Ω22+Ω2,0),0),0),0),0) - β|Lβ⊧ZFC-+∃γ|(Vγ/Vγ+1)∩Vk+1=∅(where k - inaccessible cardinal); β|Lβ⊧ZFC-+∃2nd order gap of cardinality length 1; β|Lβ⊧ZFC-+∃2-worldly cardinal C(Ω2+C(Ω2×C(Ω22+C(Ω2×C(Ω22,C(Ω22+Ω2,0),0),C(Ω22+Ω2,0),0),0),0),0) - β|Lβ⊧ZFC-+∀n∃γ|(Vγ/Vγ+1)∩Vk+n=∅(where k - inaccessible cardinal); β|Lβ⊧ZFC-+∀n∃n-worldly cardinal