Large countable ordinal notation up to Z2 and ZFC

S - Large Countable Ordinal Notation (SLCON)

This notation not well-ordered, but well-formed on KP+x,
where x - admissible ordinal or limit of admissible in this well-formed notation.
We still need well-ordered ordinal notation to get big numbers!!!
With this notation, I tried to express the existing large countable ordinals.
[here outdated and incorrect information]

Some notes:
ϒ - means pseudo-ordinal term used as diagonalizer for thing like "hyper-x"
(...0_|k|0_|k|0_|k|n) = (n)
(n_|k|0_|k|0_|k|0_|k|... m -times) = (n_|k;m|)
k - meas property of ordinal
0 (σ) - order
1 (σ'1) - inaccessibility
2 (σ'2) - mahloness
3 (σ'3) - П₃-reflecting
e.t.c
thing like 1/2{a}, 1/2{a}/3{b}{c_{d}}{{e}} means combination of this property.
zoo - means refer to Madore D., Zoo of ordinals, 2017.
[theory] - means theory of which this ordinal is model.

Up to П¹₂-TR₀:

S[σ] - ω; 1st admissible {zoo 1.4}
S[σ](1) - 1st admissible after ω; ω₁^CK [KPω], {zoo 2.1}, collapse{zoo 1.20}
S[σ](2) - 2nd admissible after ω; 1st admissible after ω₁^CK; ω₂^CK
S[σ](ω) - 1st limit of admissible; ω_ω^CK [П¹₁-CA₀], [Δ¹₂-CA₀], {zoo 2.2} collapse{zoo 1.21}
S[σ](ω+1) - 1st admissible after 1st limit of admissible; ω_ω+1^CK [KPl], [П¹₁-CA+BI]
S[σ](ω×2) - 2nd limit of admissible
S[σ](ω²) - ω-th limit of admissible
S[σ](ε₀) - ε₀-th limit of admissible; ω_ε0^CK [Δ¹₂-CA]
S[σ](S[σ](1)) - (1st admissible)-th limit of admissible;
S[σ](S[σ](ω)) - (1st limit of admissible)-th limit of admissible;
S[σ](1_|0|0) - 1st fixed point of limit of admissible [П¹₁-TR₀]
S[σ](1_|0|1) - 2nd fixed point of limit of admissible
S[σ](2_|0|0) - 1st fixed point of fixed point of limit of admissible = 1st 2-fixed point of limit of admissible
S[σ](1_|0|0_|0|0) - 1st hyper-fixed point of limit of admissible
S[σ](1_|0|0_|0|0_|0|0) - 1st (ϒ²)-order fixed point of limit of admissible
S[σ](1_|0;ω|0) - 1st (ϒ^ω)-order fixed point of limit of admissible
S[σ](1_|0;1|0|0|0) - 1st (ϒ^ϒ)-order fixed point of limit of admissible
S[σ](1_{|0;1|0;1|0|0|0|}0) - 1st (ϒ^ϒϒ)-order-fixed point of limit of admissible
S[σ'1] - 1st inaccessible = П₁-reflecting on П₂-reflecting = 1st (1st admissible after ϒ)-order-fixed point of limit of admissible = 1st admissible limit of admissible [KPi], [Δ¹₂-CA+BI], {zoo 2.3} collapse{zoo 1.22}
S[σ](S[σ'1]+1) - 1st admissible after inaccessible
S[σ'1](1) - 2nd inaccessible
S[σ'1](1_|0|0) - 1st fixed point of inaccessible
S[σ'1](1_|0|1) - 2nd fixed point of inaccessible
S[σ'1](1_|1|0) - 1st 2-inaccessible = 1st (1st admissible after ϒ)-order-fixed point of inaccessible
S[σ'1](1_|1|1) - 2nd 2-inaccessible
S[σ'1](1_|1|1_|0|0) - 1st fixed point of 2-inaccessible
S[σ'1](2_|1|0) - 1st 3-inaccessible
S[σ'1](1_|0|0_|1|0) - 1st (1st fixed point of α)-inaccessible
S[σ'1](1_|1|0_|1|0) - 1st hyper-inaccessible, [KPh], {zoo 2.4}
S[σ'1](1_|1|0_|1|1) - 2nd hyper-inaccessible
S[σ'1](1_|1|1_|1|0) - 1st 2-hyper-inaccessible
S[σ'1](2_|1|0_|1|0) - 1st hyper²-inaccessible
S[σ'1](1_|1|0_|1|0_|1|0) - 1st (ϒ²)-order-inaccessible
S[σ'1](1_|1;ω|0) - 1st (ϒ^ω)-order-inaccessible
S[σ'1](1_|1;1|1|0|0) - 1st (ϒ^ϒ)-order-inaccessible
S[σ'1](1_{|1;1|1;1|1|0|0|}0) - 1st (ϒ^ϒϒ)-order-inaccessible
S[σ'2] - 1st Mahlo = П₂-reflecting on П₂-reflecting = 1st (1st admissible after ϒ)-order-inaccessible, [KPM], {zoo 2.5} collapse{zoo 1.23}
S[σ](S[σ'2]+1) - 1st admissible after 1st Mahlo
S[σ'1](S[σ'2]+1) - 1st inaccessible after 1st Mahlo
S[σ'2](1) - 2nd Mahlo
S[σ'2](1_|0|1) - 1st fixed point of Mahlo
S[σ'2](1_|1|0) - 1st inaccessible limit of Mahlo
S[σ'2](1_|1|1) - 2nd inaccessible limit of Mahlo
S[σ'2](1_|1|1_|0|0) - 1st fixed point of inaccessible limit of Mahlo
S[σ'2](2_|1|0) - 2nd 2-inaccessible limit of Mahlo
S[σ'2](1_|0|0_|1|0) - 1st (1st fixed point of α)-inaccessible limit of Mahlo
S[σ'2](1_|1|0_|1|0) - 1st hyper-inaccessible limit of Mahlo
S[σ'2](1_|1/2|0) - 1st Mahlo limit of Mahlo
S[σ'2](1_|1/2|1) - 2nd Mahlo limit of Mahlo
S[σ'2](1_|1/2|1_|0|0) - 1st fixed point of Mahlo limit of Mahlo
S[σ'2](1_|1/2|1_|1|0) - 1st inaccessible limit of Mahlo limit of Mahlo
S[σ'2](1_|1/2|1_|1|0_|1|0) - 1st hyper-inaccessible limit of Mahlo limit of Mahlo
S[σ'2](2_|1/2|0) - 1st Mahlo limit of Mahlo limit of Mahlo = 1st Mahlo 2-limit of Mahlo
S[σ'2](1_|1/2|0_|1/2|0) - 1st Mahlo hyper-limit of Mahlo
S[σ'2](1_|2|0) - 1st 2-Mahlo
S[σ'2](1_|2|1) - 2nd 2-Mahlo
S[σ'2](1_|2|1_|0|0) - 1st fixed point of 2-Mahlo
S[σ'2](1_|2|1_|1|0) - 1st inaccessible limit of 2-Mahlo
S[σ'2](1_|2|1_|1/2|0) - 1st Mahlo limit of 2-Mahlo
S[σ'2](1_|2|1_|1/2{1}|0) - 1st 2-Mahlo limit of 2-Mahlo
S[σ'2](2_|2|0) - 1st 3-Mahlo
S[σ'2](2_|2|1_|1/2|0) - 1st Mahlo limit of 3-Mahlo
S[σ'2](2_|2|1_|1/2{1}|0) - 1st 2-Mahlo limit of 3-Mahlo
S[σ'2](2_|2|1_|1/2{2}|0) - 1st 3-Mahlo limit of 3-Mahlo
S[σ'2](β_|2|1_|1/2{γ}|0) - 1st γ-Mahlo limit of β-Mahlo
S[σ'2](1_|0|0_|2|0) - 1st (1st fixed point of α)-Mahlo
S[σ'2](1_|2|0_|2|0) - 1st hyper-Mahlo
S[σ'2](1_|2|0_|2|0_|2|0) - 1st (ϒ²)-order-Mahlo
S[σ'2](1_|2;ω|0) - 1st (ϒ^ω)-order-Mahlo
S[σ'2](1_|2;1|2|0|0) - 1st (ϒ^ϒ)-order-Mahlo
S[σ'2](1_{|2;1|2;1|2|0|0|}0) - 1st (ϒ^ϒϒ)-order-Mahlo
S[σ'3] - 1st П₃-reflecting = 1st (1st admissible after ϒ)-order-Mahlo, [KP+П₃-ref], {zoo 2.6} collapse{zoo 1.24}
S[σ'3](1) - 2nd П₃-reflecting
S[σ'3](1_|0|0) - 1st fixed point of П₃-reflecting
S[σ'3](1_|1|0) - 1st inaccessible limit of П₃-reflecting
S[σ'3](1_|1/2|0) - 1st Mahlo limit of П₃-reflecting
S[σ'3](1_|1/2{α}|0) - 1st α-Mahlo limit of П₃-reflecting
S[σ'3](1_|1/3|0) - 1st П₃-reflecting limit of П₃-reflecting
S[σ'3](1_|2|0) - 1st Mahlo in which П₃-reflecting are stationary = 1st П₃-reflecting (1st admissible after ϒ)-limit of Mahlo
S[σ'3](1_|2|1_|1/3|0) - 1st П₃-reflecting limit of Mahlo in which П₃-reflecting are stationary
S[σ'3](1_|2|1_|1/2/3|0) - 1st Mahlo in which П₃-reflecting are stationary limit of Mahlo in which П₃-reflecting are stationary
S[σ'3](2_|2|0) - 1st 2-Mahlo in which П₃-reflecting are stationary
S[σ'3](2_|2|1_|1/3|0) - 1st П₃-reflecting limit of 2-Mahlo in which П₃-reflecting are stationary
S[σ'3](2_|2|1_|1/2/3|0) - 1st Mahlo in which П₃-reflecting are stationary limit of 2-Mahlo in which П₃-reflecting are stationary
S[σ'3](2_|2|1_|1/2{1}/3|0) - 1st 2-Mahlo in which П₃-reflecting are stationary limit of 2-Mahlo in which П₃-reflecting are stationary
S[σ'3](1_|2/3|0) - 1st П₃-reflecting in which П₃-reflecting are stationary = 1st (1st admissible after ϒ)-order-Mahlo in which П₃-reflecting are stationary
S[σ'3](1_|2/3|1_|1/3|0) - 1st П₃-reflecting limit of П₃-reflecting in which П₃-reflecting are stationary
S[σ'3](1_|2/3|1_|1/2/3|0) - 1st Mahlo in which П₃-reflecting are stationary limit of П₃-reflecting in which П₃-reflecting are stationary
S[σ'3](1_|2/3|1_|1/2{α}/3|0) - 1st α-Mahlo in which П₃-reflecting are stationary limit of П₃-reflecting in which П₃-reflecting are stationary
S[σ'3](1_|2/3|1_|1/3{1}|0) - 1st П₃-reflecting in which П₃-reflecting are stationary limit of П₃-reflecting in which П₃-reflecting are stationary
S[σ'3](1_|2/3|1_|2|0) - 1st Mahlo in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary
S[σ'3](1_|2/3|1_|2|1_|1/2/3{1}|0) - 1st Mahlo in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary limit of Mahlo in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary
S[σ'3](1_|2/3|2_|2|0) - 1st 2-Mahlo in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary
S[σ'3](1_|2/3|2_|2|1_{|1/2{1}/3{1}|}0) - 1st 2-Mahlo in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary limit of 2-Mahlo in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary
S[σ'3](2_|2/3|) - 1st П₃-reflecting in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary = 1st П₃-reflecting in which П₃-reflecting are 2-stationary
S[σ'3](2_|2/3|1_|1/3|0) - 1st П₃-reflecting limit of П₃-reflecting in which П₃-reflecting are 2-stationary
S[σ'3](2_|2/3|1_|1/2{α}/3|0) - 1st Mahlo in which П₃-reflecting are stationary limit of П₃-reflecting in which П₃-reflecting are 2-stationary
S[σ'3](2_|2/3|1_|1/3{1}|0) - 1st П₃-reflecting in which П₃-reflecting are stationary limit of П₃-reflecting in which П₃-reflecting are 2-stationary
S[σ'3](2_|2/3|1_{|1/2{α}/3{1}|}0) - 1st Mahlo in which (П₃-reflecting in which П₃-reflecting are stationary) are stationary limit of П₃-reflecting in which П₃-reflecting are 2-stationary
S[σ'3](3_|2/3|) - 1st П₃-reflecting in which П₃-reflecting are 3-stationary
S[σ'3](1_|2/3|0_|2/3|0) - 1st П₃-reflecting in which П₃-reflecting are hyper-stationary
S[σ'3](1_|3|) - 1st П₃-reflecting onto П₃-reflecting = 1st 2-П₃-reflecting = 1st П₃-reflecting in which П₃-reflecting are (1st admissible after ϒ)-order-stationary
S[σ'3](1_|3|1) - 2nd 2-П₃-reflecting
S[σ'3](1_|3|1_|0|0) - 1st fixed point of 2-П₃-reflecting
S[σ'3](1_|3|1_|1|0) - 1st inaccessible limit of 2-П₃-reflecting
S[σ'3](1_|3|1_|1/2|0) - 1st Mahlo limit of 2-П₃-reflecting
S[σ'3](1_|3|1_|1/2{α}|0) - 1st α-Mahlo limit of 2-П₃-reflecting
S[σ'3](1_|3|1_|1/3|0) - 1st П₃-reflecting limit of 2-П₃-reflecting
S[σ'3](1_|3|1_|1/2/3|0) - 1st Mahlo in which П₃-reflecting are stationary limit of 2-П₃-reflecting
S[σ'3](1_|3|1_|1/2{α}/3|0) - 1st α-Mahlo in which П₃-reflecting are stationary limit of 2-П₃-reflecting
S[σ'3](1_|3|1_|1/3{α}|0) - 1st П₃-reflecting in which П₃-reflecting are α-stationary limit of 2-П₃-reflecting
S[σ'3](1_|3|1_1/3{{1}}|0) - 1st 2-П₃-reflecting limit of 2-П₃-reflecting
S[σ'3](1_|3|1_|2|0) - 1st Mahlo in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|1_|2|1_|1/2/3{{1}}|0) - 1st Mahlo in which 2-П₃-reflecting are stationary limit of Mahlo in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|2_|2|0) - 1st 2-Mahlo in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|2_|2|1_{|1/2{1}/3{{1}}|}0) - 1st 2-Mahlo in which 2-П₃-reflecting are stationary limit of 2-Mahlo in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|1_|2/3|) - 1st П₃-reflecting in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|1_|2/3|1_{|1/3{1}{{1}}|}0) - 1st П₃-reflecting in which 2-П₃-reflecting are stationary limit of П₃-reflecting in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|1_|2/3{1}|0) - 1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|1_|2/3{1}|1_{|1/3{1{1}}{{1}}|}0) - 1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary limit of 2-П₃-reflecting in which 2-П₃-reflecting are stationary
S[σ'3](1_|3|1_|2/3{1}|1_|2/3|0) - 1st П₃-reflecting in which (1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary) are stationary
S[σ'3](1_|3|1_|2/3{1}|1_|2/3|1_{|1/3{1}{1{1}}{{1}}|}0) - 1st П₃-reflecting in which (1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary) are stationary limit of П₃-reflecting in which (1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary) are stationary
S[σ'3](1_|3|1_|2/3{1}|1_|2/3|1_|2|0) - 1st Mahlo in which (1st П₃-reflecting in which (1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary) are stationary) are stationary
S[σ'3](1_|3|1_|2/3{1}|1_|2/3|1_|2|1_{|1/2{1}/3{1}{1{1}}{{1}}|}0) - 1st Mahlo in which (1st П₃-reflecting in which (1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary) are stationary) are stationary limit of Mahlo in which (1st П₃-reflecting in which (1st 2-П₃-reflecting in which 2-П₃-reflecting are stationary) are stationary) are stationary
S[σ'3](1_|3|2_|2/3{1}|0) - 1st 2-П₃-reflecting in which 2-П₃-reflecting are 2-stationary
S[σ'3](1_|3|2_|2/3{1}|1_{|1/3{2{1}}{{1}}|}0) - 1st 2-П₃-reflecting in which 2-П₃-reflecting are 2-stationary limit of 2-П₃-reflecting in which 2-П₃-reflecting are 2-stationary
S[σ'3](2_|3|0) - 1st 3-П₃-reflecting
S[σ'3](2_|3|1_|2/3|0) - 1st П₃-reflecting in which 3-П₃-reflecting are stationary
S[σ'3](2_|3|1_|2/3{1}|0) - 1st 2-П₃-reflecting in which 3-П₃-reflecting are stationary
S[σ'3](2_|3|1_|2/3{1}|1_{|1/3{1{1}}{{2}}|}0) - 1st П₃-reflecting in which 3-П₃-reflecting are stationary limit of 3-П₃-reflecting in which 3-П₃-reflecting are stationary
S[σ'3](2_|3|1_|2/3{2}|0) - 1st 3-П₃-reflecting in which 3-П₃-reflecting are stationary
S[σ'3](2_|3|1_|2/3{2}|1_{|1/3{1}{{2}}|}0) - 1st П₃-reflecting in which 3-П₃-reflecting are stationary limit of 3-П₃-reflecting in which 3-П₃-reflecting are stationary
S[σ'3](2_|3|1_|2/3{2}|1_{|1/3{1{1}}{{2}}|}0) - 1st 2-П₃-reflecting in which 3-П₃-reflecting are stationary limit of 3-П₃-reflecting in which 3-П₃-reflecting are stationary
S[σ'3](2_|3|1_|2/3{2}|1_{|1/3{1{2}}{{2}}|}0) - 1st 3-П₃-reflecting in which 3-П₃-reflecting are stationary limit of 3-П₃-reflecting in which 3-П₃-reflecting are stationary
S[σ'3](2_|3|1_|2/3{2}|1_|2/3{1}|1_|2/3|0) - 1st П₃-reflecting in which (1st 2-П₃-reflecting in which (1st 3-П₃-reflecting in which 3-П₃-reflecting are stationary) are stationary) are stationary
S[σ'3](2_|3|1_|2/3{2}|1_|2/3{1}|1_|2/3|1_{|1/3{1}{1{1}}{1{2}}{{2}}|}0) - 1st П₃-reflecting in which (1st 2-П₃-reflecting in which (1st 3-П₃-reflecting in which 3-П₃-reflecting are stationary) are stationary) are stationary limit of П₃-reflecting in which (1st 2-П₃-reflecting in which (1st 3-П₃-reflecting in which 3-П₃-reflecting are stationary) are stationary) are stationary
S[σ'3](1_|0|0_|3|0) - 1st (1st fixed point of α)-П₃-reflecting
S[σ'3](1_|3|0_|3|0) - 1st hyper-П₃-reflecting
S[σ'3](1_|3|0_|3|0_|3|0) - 1st (ϒ²)-order-П₃-reflecting
S[σ'3](1_|3;ω|0) - 1st (ϒ^ω)-order--П₃-reflecting
S[σ'3](1_|3;1|3|0|0) - 1st (ϒ^ϒ)-order-П₃-reflecting
S[σ'3](1_{|3;1|3;1|3|0|0|}0) - 1st (ϒ^ϒϒ)-order-П₃-reflecting
S[σ'4] - 1st П₄-reflecting = 1st (1st admissible after ϒ)-order-П₃-reflecting, [KP+П₄-ref]
S[σ'4](1_|0|0) - 1st fixed point of П₄-reflecting
S[σ'4](1_|1|0) - 1st inaccessible limit of П₄-reflecting
S[σ'4](1_|1/2|0) - 1st Mahlo limit of П₄-reflecting
S[σ'4](1_|1/2{α}|0) - 1st α-Mahlo limit of П₄-reflecting
S[σ'4](1_|1/3|0) - 1st П₃-reflecting limit of П₄-reflecting
S[σ'4](1_|1/3{α}|0) - 1st П₃-reflecting in which П₃-reflecting are α-stationary limit of П₄-reflecting
S[σ'4](1_|1/3{{α}}|0) - 1st α-П₃-reflecting limit of П₄-reflecting
S[σ'4](1_{|1/3{β{γ}}{{α}}|}0) - 1st γ-П₃-reflecting in which α-П₃-reflecting are β-stationary limit of П₄-reflecting
S[σ'4](1_|1/4|0) - 1st П₄-reflecting limit of П₄-reflecting
S[σ'4](1_|2|0) - 1st Mahlo in which П₄-reflecting are stationary
S[σ'4](1_|2|1_|1/4{1}|0) - 1st Mahlo in which П₄-reflecting are stationary limit of Mahlo in which П₄-reflecting are stationary
S[σ'4](1_|2/3{α}|0) - 1st α-П₃-reflecting in which П₄-reflecting are stationary
S[σ'4](1_|2/4|0) - 1st П₄-reflecting in which П₄-reflecting are stationary
S[σ'4](1_|3|0) - 1st П₃-reflecting that is П₃-reflecting onto П₄-reflecting
S[σ'4](1_|3|1_|1/4{{1}}|0) - 1st П₃-reflecting that is П₃-reflecting onto П₄-reflecting limit of П₃-reflecting that is П₃-reflecting onto П₄-reflecting
S[σ'4](2_|3|0) - 1st П₃-reflecting that is 2-П₃-reflecting onto П₄-reflecting
S[σ'4](1_|3/4|0) - 1st П₄-reflecting that is П₃-reflecting onto П₄-reflecting
S[σ'4](2_|3/4|0) - 1st П₄-reflecting that is 2-П₃-reflecting onto П₄-reflecting
S[σ'4](1_|4|0) - 1st П₄-reflecting that is П₄-reflecting onto П₄-reflecting = 1st 2-П₄-reflecting
S[σ'4](1_|4|1_|1/4|0) - 1st П₄-reflecting limit of 2-П₄-reflecting
S[σ'4](1_|4|1_|1/2{α}/4|0) - 1st Mahlo in witch П₄-reflecting are stationary limit of 2-П₄-reflecting
S[σ'4](1_|4|1_|1/3{α}/4|0) - 1st П₃-reflecting in witch П₄-reflecting are α-stationary limit of 2-П₄-reflecting
S[σ'4](1_|4|1_|1/4{α}|0) - 1st П₄-reflecting in which П₄-reflecting are α-stationary limit of 2-П₄-reflecting
S[σ'4](1_|4|1_{|1/3{{α}}/4|}0) - 1st α-П₃-reflecting that is П₃-reflecting onto П₄-reflecting limit of 2-П₄-reflecting
S[σ'4](1_|4|1_{|1/3{β{γ}}{{α}}/4|}0) - 1st γ-П₃-reflecting in which α-П₃-reflecting are β-stationary that is П₃-reflecting onto П₄-reflecting limit of 2-П₄-reflecting
S[σ'4](1_|4|1_|1/4{{α}}|0) - 1st П₄-reflecting that is α-П₃-reflecting onto П₄-reflecting limit of 2-П₄-reflecting
S[σ'4](1_|4|1_{|1/4{β{γ}}{{α}}|}0) - 1st γ-П₃-reflecting in which П₄-reflecting are β-stationary that is α-П₃-reflecting onto П₄-reflecting limit of 2-П₄-reflecting
S[σ'4](1_|4|1_|1/4{{{1}}}|0) - 1st 2-П₄-reflecting limit of 2-П₄-reflecting
S[σ'4](1_|4|1_|2/4|0) - 1st П₄-reflecting in which 2-П₄-reflecting are stationary
S[σ'4](1_|4|1_|2/3{α}/4|0) - 1st α-П₃-reflecting that is П₃-reflecting onto П₄-reflecting in which 2-П₄-reflecting are stationary
S[σ'4](1_|4|1_|2/4{α}|0) - 1st П₄-reflecting that is α-П₃-reflecting onto П₄-reflecting in which 2-П₄-reflecting are stationary
S[σ'4](1_|4|1_|2/4{{1}}|0) - 1st 2-П₄-reflecting in which 2-П₄-reflecting are stationary
S[σ'4](1_|4|1_|3/4|0) - 1st П₄-reflecting that is П₃-reflecting onto 2-П₄-reflecting
S[σ'4](1_|4|2_|3/4|0) - 1st П₄-reflecting that is 2-П₃-reflecting onto 2-П₄-reflecting
S[σ'4](1_|4|1_|3/4{1}|0) - 1st 2-П₄-reflecting that is П₃-reflecting onto 2-П₄-reflecting
S[σ'5] - 1st П₅-reflecting = 1st (1st admissible after ϒ)-order-П₄-reflecting, [KP+П₅-ref]
S[σ'5](1_|0|0) - 1st fixed point of П₅-reflecting
S[σ'5](1_|1|0) - 1st inaccessible limit of П₄-reflecting
S[σ'5](1_|1/2|0) - 1st Mahlo limit of П₅-reflecting
S[σ'5](1_|1/3|0) - 1st П₃-reflecting limit of П₅-reflecting
S[σ'5](1_|1/4|0) - 1st П₄-reflecting limit of П₅-reflecting
S[σ'5](1_|1/5|0) - 1st П₄-reflecting limit of П₅-reflecting
S[σ'5](1_|2|0) - 1st Mahlo in which П₅-reflecting are stationary
S[σ'5](1_|2/3|0) - 1st П₃-reflecting in which П₅-reflecting are stationary
S[σ'5](1_|2/4|0) - 1st П₄-reflecting in which П₅-reflecting are stationary
S[σ'5](1_|2/5|0) - 1st П₅-reflecting in which П₅-reflecting are stationary
S[σ'5](1_|3|0) - 1st П₃-reflecting that is П₃-reflecting onto П₅-reflecting
S[σ'5](1_|3/4|0) - 1st П₄-reflecting that is П₃-reflecting onto П₅-reflecting
S[σ'5](1_|3/5|0) - 1st П₅-reflecting that is П₃-reflecting onto П₅-reflecting
S[σ'5](1_|4|0) - 1st П₄-reflecting that is П₄-reflecting onto П₅-reflecting
S[σ'5](1_|4/5|0) - 1st П₅-reflecting that is П₄-reflecting onto П₅-reflecting
S[σ'5](1_|5|0) - 1st П₅-reflecting that is П₅-reflecting onto П₅-reflecting = 2-П₅-reflecting
S[σ'n] - 1st П_n-reflecting, [KP+П_n-ref]
S[σ'ω] = S[σ+1] - 1st (+1)-stable; L_σ≺₁L_σ+1, [KP+П_ω-ref], {zoo 2.7} collapse{zoo 1.25}
S[σ'ω](1_|0|0) = S[σ+1](1_|σ|0) - 1st fixed point of (+1)-stable
S[σ'ω](1_|1|0) = S[σ+1](1_|σ'1|0) - 1st inaccessible limit of (+1)-stable
S[σ'ω](1_|1/2|0) = S[σ+1](1_|σ'1/σ'2|0) - 1st Mahlo limit of (+1)-stable
S[σ'ω](1_|1/3|0) = S[σ+1](1_|σ'1/σ'3|0) - 1st П₃-reflecting limit of (+1)-stable
S[σ'ω](1_|1/ω|0) = S[σ+1](1_|σ'1/σ+1|0) - 1st (+1)-stable limit of (+1)-stable
S[σ'ω](1_|2|0) = S[σ+1](1_|σ'2|0) - 1st Mahlo in which (+1)-stable are stationary
S[σ'ω](1_|2/3|0) = S[σ+1](1_|σ'2/σ'3|0) - 1st П₃-reflecting in which (+1)-stable are stationary
S[σ'ω](1_|2/4|0) = S[σ+1](1_|σ'2/σ'4|0) - 1st П₄-reflecting in which (+1)-stable are stationary
S[σ'ω](1_|2/ω|0) = S[σ+1](1_|σ'2/σ+1|0) - 1st (+1)-stable in which (+1)-stable are stationary
S[σ'ω](1_|3|0) = S[σ+1](1_|σ'3|0) - 1st П₃-reflecting that is П₃-reflecting onto (+1)-stable
S[σ'ω](1_|3/4|0) = S[σ+1](1_|σ'3/σ'4|0) - 1st П₄-reflecting that is П₃-reflecting onto (+1)-stable
S[σ'ω](1_|3/5|0) = S[σ+1](1_|σ'3/σ'5|0) - 1st П₅-reflecting that is П₃-reflecting onto (+1)-stable
S[σ'ω](1_|3/ω|0) = S[σ+1](1_|σ'3/σ+1|0) - 1st (+1)-stable that is П₃-reflecting onto (+1)-stable
S[σ'ω](1_|ω|0) = S[σ+1](1_|σ+1|0) - 1st 2-(+1)-stable
S[σ'ω](1_|ω|1_|1/ω{α}|0) = S[σ+1](1_|σ+1|1_{|σ'1/σ+1{α}^|σ'2||}0) - 1st (+1)-stable in which (+1)-stable are α-stationary limit of 2-(+1)-stable
S[σ'ω](1_|ω|1_|1/ω{{α}}|0) = S[σ+1](1_|σ+1|1_{|σ'1/σ+1{α}^|σ'3||}0) - 1st (+1)-stable that is α-П₃-reflecting onto (+1)-stable limit of 2-(+1)-stable
S[σ'ω](1_|ω|1_{|1/ω{{{α}}}|}0) = S[σ+1](1_|σ+1|1_{|σ'1/σ+1{α}^|σ'4||}0) - 1st (+1)-stable that is α-П₄-reflecting onto (+1)-stable limit of 2-(+1)-stable
S[σ'ω](1_|ω|1_{|1/ω{1}^|ω||}0) = S[σ+1](1_|σ+1|1_{|σ'1/σ+1{1}^|σ+1||}0) - 1st (+1)-stable that is α-П₄-reflecting onto (+1)-stable limit of 2-(+1)-stable
S[σ'ω](1_|ω|1_|2/ω{α}|0) = S[σ+1](1_|σ+1|1_{|σ'2/σ+1{α}^|σ'3||}0) - 1st (+1)-stable that is α-П₃-reflecting onto (+1)-stable in which 2-(+1)-stable are stationary
S[σ'ω](1_|ω|1_|2/ω{{α}}|0) = S[σ+1](1_|σ+1|1_{|σ'2/σ+1{α}^|σ'4||}0) - 1st (+1)-stable that is α-П₄-reflecting onto (+1)-stable in which 2-(+1)-stable are stationary
S[σ'ω](1_|ω|1_{|2/ω{{{α}}}|}0) = S[σ+1](1_|σ+1|1_{|σ'2/σ+1{α}^|σ'5||}0) - 1st (+1)-stable that is α-П₅-reflecting onto (+1)-stable in which 2-(+1)-stable are stationary
S[σ'ω](1_|ω|1_{|2/ω{1}^|ω||}0) = S[σ+1](1_|σ+1|1_{|σ'2/σ+1{1}^|σ+1||}0) - 1st (+1)-stable that is (+1)-stable onto (+1)-stable in which 2-(+1)-stable are stationary
S[σ'ω](1_|ω|1_|3/ω{α}|0) = S[σ+1](1_|σ+1|1_{|σ'3/σ+1{α}^|σ'4||}0) - 1st (+1)-stable that is α-П₄-reflecting onto (+1)-stable that is П₃-reflecting onto 2-П₄-reflecting
S[σ'ω](1_|ω|1_|3/ω{{α}}|0) = S[σ+1](1_|σ+1|1_{|σ'3/σ+1{α}^|σ'5||}0) - 1st (+1)-stable that is α-П₅-reflecting onto (+1)-stable that is П₃-reflecting onto 2-П₄-reflecting
S[σ'ω](1_|ω|1_{|3/ω{{{α}}}|}0) = S[σ+1](1_|σ+1|1_{|σ'3/σ+1{α}^|σ'6||}0) - 1st (+1)-stable that is α-П₆-reflecting onto (+1)-stable that is П₃-reflecting onto 2-П₄-reflecting
S[σ'ω](1_|ω|1_{|3/ω{1}^|ω||}0) = S[σ+1](1_|σ+1|1_{|σ'3/σ+1{1}^|σ+1||}0) - 1st (+1)-stable that is (+1)-stable onto (+1)-stable that is П₃-reflecting onto 2-П₄-reflecting
S[σ'ω](1_|ω|0_|ω|0) = S[σ+1](1_|σ+1|0_|σ+1|0) - 1st hyper-(+1)-stable
S[σ'ω](1_|ω|0_|ω|0_|ω|0) = S[σ+1](1_|σ+1|0_|σ+1|0_|σ+1|0) - 1st (ϒ²)-order-(+1)-stable
S[σ'ω](1_|ω;ω|0) = S[σ+1](1_|σ+1;ω|0) - 1st (ϒ^ω)-order-(+1)-stable
S[σ'ω](1_|ω;1|ω|0|0) = S[σ+1](1_{|σ+1;1|σ+1|0|}0) - 1st (ϒ^ϒ)-order-(+1)-stable
S[σ'ω+1] = S[σ+1'1] - 1st (+1)-П₁-reflecting (L_σ+1⊧φ→∃β<σ(L_β+1⊧φ); φ is П₁-formula); 1st (1st admissible after ϒ)-order-(+1)-stable
S[σ'ω+1](1_|1/ω+1|0) = S[σ+1'1](1_{|σ'1/σ+1'1|}0) - 1st (+1)-П₁-reflecting limit of (+1)-П₁-reflecting
S[σ'ω+1](1_|2/ω+1|0) = S[σ+1'1](1_{|σ'2/σ+1'1|}0) - 1st (+1)-П₁-reflecting in which (+1)-П₁-reflecting are stationary
S[σ'ω+1](1_|3/ω+1|0) = S[σ+1'1](1_{|σ'3/σ+1'1|}0) - 1st (+1)-П₁-reflecting that is П₃-reflecting onto (+1)-П₁-reflecting
S[σ'ω+1](1_|ω/ω+1|0) = S[σ+1'1](1_{|σ+1/σ+1'1|}0) - 1st (+1)-П₁-reflecting that is (+1)-stable onto (+1)-П₁-reflecting
S[σ'ω+1](1_|ω+1|0) = S[σ+1'1](1_|σ+1'1|0) - 1st 2-(+1)-П₁-reflecting
S[σ'ω+2] = S[σ+1'2] - 1st (+1)-П₂-reflecting (L_σ+1⊧φ→∃β<σ(L_β+1⊧φ); φ is П₂-formula)
S[σ'ω+n] = S[σ+1'n] - 1st (+1)-П_n-reflecting (L_σ+1⊧φ→∃β<σ(L_β+1⊧φ); φ is П_n-formula)
S[σ'ω×2] = S[σ+2] - 1st (+2)-stable; L_σ≺₁L_σ+2
S[σ'ω×2+1] = S[σ+2'1] - 1st (+2)-П₁-reflecting (L_σ+2⊧φ→∃β<σ(L_β+2⊧φ); φ is П₁-formula)
S[σ'ω×2+2] = S[σ+2'2] - 1st (+2)-П₂-reflecting (L_σ+2⊧φ→∃β<σ(L_β+2⊧φ); φ is П₂-formula)
S[σ'ω×2+n] = S[σ+2'n] -1st (+2)-П_n-reflecting (L_σ+2⊧φ→∃β<σ(L_β+2⊧φ); φ is П_n-formula)
S[σ'ω×3] = S[σ+3] - 1st (+3)-stable; L_σ≺₁L_σ+3
S[σ'ω²] = S[σ+ω] = S[σ+S[σ]] - 1st (+ω)-stable; L_σ≺₁L_σ+ω
S[σ'ω³] = S[σ+ω×2] - 1st (+ω×2)-stable; L_σ≺₁L_σ+ω×2
S[σ'ω^ω] = S[σ+ω²] - 1st (+ω²)-stable; L_σ≺₁L_σ+ω²
S[σ'ω^ωω] = S[σ+ω^ω] - 1st (+ω^ω)-stable; L_σ≺₁L_σ+ω^ω
S[σ'ε₀] = S[σ+ε₀] - 1st (+ε₀)-stable; L_σ≺₁L_σ+ε0
S[σ'S[σ](1)] = S[σ+S[σ](1)] - 1st (+ω₁^CK)-stable; L_σ≺₁L_ω1^CK
S[σ'S[σ'ω]] = S[σ+S[σ+1]] - 1st (+(+1)-stable)-stable; L_σ≺₁L_{σ+Lβ≺Lβ+1}
S[σ+α] - 1st (+α)-stable; L_σ≺₁L_σ+α
S[σ×2] = S[σ+σ] - 1st σ=(+σ)-stable; L_σ≺₁L_σ+σ [KPi+∀n∃σ≥n(L_σ≺₁L_σ+n)], collapse{zoo 1.26}
S[σ×2+1] - 1st σ=(+σ+1)-stable
S[σ×2+α] - 1st σ=(+σ+α)-stable
S[σ×3] - 1st σ=(+σ×2)-stable
S[σ×ω] - 1st σ=(+σ×ω)-stable
S[σ×α] - 1st σ=(+σ×α)-stable
S[σ²] - 1st σ=(+σ×σ)-stable
S[σ^ω] - 1st σ=(+σ^ω)-stable
S[σ^α] - 1st σ=(+σ^α)-stable
S[σ^σ] - 1st σ=(+σ^σ)-stable
S[ε_σ+1] - 1st σ=(+ε_σ+1)-stable
S[S₂[σ₂](σ+1)] - 1st σ=(1st admissible after σ)-stable; 1st σ=(σ⁺)-stable; 1st σ=(next admissible)-stable [KP+П¹₁-ref], {zoo 2.8}
S[S₂[σ₂](σ+1)](1_{|σ'1/S2[σ2](σ+1)|}) - 1st σ=(σ⁺)-stable limit of σ=(σ⁺)-stable
S[S₂[σ₂](σ+1)](1_{|σ'2/S2[σ2](σ+1)|}) - 1st σ=(σ⁺)-stable in which σ=(σ⁺)-stable are stationary
S[S₂[σ₂](σ+1)](1_{|σ'3/S2[σ2](σ+1)|}) - 1st σ=(σ⁺)-stable that is П₃-reflecting onto σ=(σ⁺)-stable
S[S₂[σ₂](σ+1)](1_{|σ+1/S2[σ2](σ+1)|}) - 1st σ=(σ⁺)-stable that is (+1)-stable onto σ=(σ⁺)-stable
S[S₂[σ₂](σ+1)](1_{|σ+σ/S2[σ2](σ+1)|}) -  1st σ=(σ⁺)-stable that is σ=(+σ)-stable onto σ=(σ⁺)-stable
S[S₂[σ₂](σ+1)](1_{|S2[σ2](σ+1)|}) - 1st 2-σ=(σ⁺)-stable
S[S₂[σ₂](σ+1)'1] - 1st σ=(σ⁺)-П₁-reflecting
S[S₂[σ₂](σ+1)+1] - 1st σ=(σ⁺+1)-stable
S[S₂[σ₂](σ+1)+α] - 1st σ=(σ⁺+α)-stable
S[S₂[σ₂](σ+1)+σ] - 1st σ=(σ⁺+σ)-stable
S[S₂[σ₂](σ+1)+S₂[σ₂](σ+1)] - 1st σ=(σ⁺×2)-stable
S[S₂[σ₂](σ+1)^{S₂[σ₂](σ+1)}] - 1st σ=(σ^+σ+)-stable
S[ε_{S2[σ2](σ+1)}] - 1st σ=(ε_σ⁺+1)-stable
S[S₂[σ₂](σ+2)] - 1st σ=(1st admissible after σ⁺)-stable; 1st σ=(σ⁺⁺)-stable; 1st σ=(next 2nd admissible)-stable {zoo 2.10}
S[S₂[σ₂](σ+2)](1_{|σ'1/S2[σ2](σ+2)|}) - 1st σ=(σ⁺⁺)-stable limit of σ=(σ⁺⁺)-stable
S[S₂[σ₂](σ+2)](1_{|σ'2/S2[σ2](σ+2)|}) - 1st σ=(σ⁺⁺)-stable in which σ=(σ⁺⁺)-stable are stationary
S[S₂[σ₂](σ+2)](1_{|σ'3/S2[σ2](σ+2)|}) - 1st σ=(σ⁺⁺)-stable that is П₃-reflecting onto σ=(σ⁺⁺)-stable
S[S₂[σ₂](σ+2)](1_{|σ+1/S2[σ2](σ+2)|}) - 1st σ=(σ⁺⁺)-stable that is (+1)-stable onto σ=(σ⁺⁺)-stable
S[S₂[σ₂](σ+2)](1_{|σ+σ/S2[σ2](σ+2)|}) - 1st σ=(σ⁺⁺)-stable that is σ=(+σ)-stable onto σ=(σ⁺⁺)-stable
S[S₂[σ₂](σ+2)](1_{|S2[σ2](σ+1)/S2[σ2](σ+2)|}) - 1st (σ⁺⁺)-stable that is σ=(σ⁺)-stable onto σ=(σ⁺⁺)-stable
S[S₂[σ₂](σ+2)](1_{|S2[σ2](σ+2)|}) - 1st 2-σ=(σ⁺⁺)-stable
S[S₂[σ₂](σ+3)] - 1st σ=(1st admissible after σ⁺⁺)-stable; 1st σ=(σ⁺⁺⁺)-stable; 1st σ=(next 3d admissible)-stable
S[S₂[σ₂](σ+ω)] - 1st σ=(next ω-th admissible)-stable [limit of DAN]
S[S₂[σ₂](σ+α)] - 1st σ=(next α-th admissible)-stable
S[S₂[σ₂](σ+σ)] - 1st σ=(next σ-th admissible)-stable
S[S₂[σ₂](σ^σ)] - 1st σ=(next σ^σ-th admissible)-stable
S[S₂[σ₂](S₂[σ₂](σ+1))] - 1st σ=(next σ⁺-th admissible)-stable; σ=(next next admissible)-stable
S[S₂[σ₂](S₂[σ₂](σ+2))] - 1st σ=(next σ⁺⁺-th admissible)-stable; σ=(next next 2nd admissible)-stable
S[S₂[σ₂](S₂[σ₂](S₂[σ₂](σ+1)))] - 1st σ=(next next next admissible)-stable
S[S₂[σ₂](1_|σ|σ+1)] - 1st σ=(1st limit of α-next admissible)-stable
S[S₂[σ₂]([S₂[σ₂](1_|σ|σ+1)+1)] - 1st σ=(next after 1st limit of α-next admissible)-stable
S[S₂[σ₂]([S₂[σ₂](1_|σ|σ+1)+2)] - 1st σ=(next 2nd after 1st limit of α-next admissible)-stable
S[S₂[σ₂](S₂[σ₂]([S₂[σ₂](1_|σ|σ+1)+1))] - 1st σ=(next next after 1st limit of α-next admissible)-stable
S[S₂[σ₂](1_|σ|σ+2)] - 1st σ=(2nd limit of α-next admissible)-stable
S[S₂[σ₂](1_|σ|σ+σ)] - 1st σ=(σ-th limit of α-next admissible)-stable
S[S₂[σ₂](2_|σ|σ+1)] - 1st σ=(1st fixed point of limit of α-next admissible)-stable
S[S₂[σ₂](2_|σ|σ+2)] - 1st σ=(2nd fixed point of limit of α-next admissible)-stable
S[S₂[σ₂](2_|σ|σ+σ)] - 1st σ=(σ-th fixed point of limit of α-next admissible)-stable
S[S₂[σ₂](2_|σ|σ+2)] - 1st σ=(1st 2-fixed point of limit of α-next admissible)-stable
S[S₂[σ₂](σ_|σ|1)] - 1st σ=(1st σ-fixed point of limit of α-next admissible)-stable
S[S₂[σ₂](1_|σ|0_|σ|σ+1)] - 1st σ=(1st hyper-fixed point of limit α-next admissible)-stable
S[S₂[σ₂](1_|σ'1|σ+1)] - 1st σ=(1st inaccessible limit of limit of α-next admissible)-stable; 1st σ=(1st П₂-reflecting on α-next admissible)-stable
S[S₂[σ₂](1_|σ'2|σ+1)] - 1st σ=(1st Mahlo limit on fixed point of α-next admissible)-stable; 1st σ=(1st П₂-reflecting on П₂-reflecting on α-next admissible)-stable
S[S₂[σ₂](1_|σ'3|σ+1)] - 1st σ=(1st П₃-reflecting on α-next admissible)-stable
S[S₂[σ₂](1_|σ+1|σ+1)] - 1st σ=(1st (+1)-stable on α-next admissible)-stable
S[S₂[σ₂](1_{|S2[σ2](1σ+1)|}σ+1)] - 1st σ=(1s (next admissible)-stable on α-next admissible)-stable
S[S₂[σ₂](1_{|S2[σ2](1|σ|σ+1)|}σ+1)] - 1st σ=(1s (1st limit α-next admissible)-stable on α-next admissible)-stable;
S[S₂[σ₂](1_|α|σ+1)] = S[α↦S₂[σ₂](1_|α|σ+1)] - 1st σ=(1st limit on limit of α-next admissible)-stable = 1st σ=(1st 2-limit of next admissible)-stable
S[S₂[σ₂](1_|α|σ+2)] - 1st σ=(2nd 2-limit of next admissible)-stable
S[S₂[σ₂](2_|α|σ+1)] - 1st σ=(1st 3-limit of next admissible)-stable
S[S₂[σ₂](1_|α|0_|α|σ+1)] - 1st σ=(hyper-limit of next admissible)-stable
S[S₂[σ₂](1_|α|0_|α|0_|α|σ+1)] - 1st σ=((ϒ^ω)-order-limit of next admissible)-stable
S[S₂[σ₂](1_|σ2|σ+1)] - 1st σ=(σ-next admissible)-stable
S[S₂[σ₂](1_|σ2|σ+2)] - 1st σ=(σ-next 2nd admissible)-stable
S[S₂[σ₂](1_|σ2|S[σ₂](1_|σ2|σ+1))] - 1st σ=(σ+1-next admissible)-stable
S[S₂[σ₂](1_|σ2|1_|σ|σ+1)] - 1st σ=(1st limit of σ+α-next admissible)-stable
S[S₂[σ₂](2_|σ2|σ+1)] - 1st σ=(σ+σ-next admissible)-stable
S[S₂[σ₂](ω_|σ2|σ+1)] - 1st σ=(σ×ω-next admissible)-stable
S[S₂[σ₂](S[S₂[σ₂](σ+1)]_|σ2|1)] - 1st σ=(σ=(next admissible)-stable-next admissible)-stable
S[S₂[σ₂](σ_|σ2|1)] - 1st σ=((σ level of σ)-next admissible)-stable
S[S₂[σ₂](1_|σ2|0_|σ2|σ+1)] - 1st σ=((hyper level of σ)-next admissible)-stable
S[S₂[σ₂'1](σ+1)] - 1st σ=(next inaccessible)-stable {zoo 2.11}
S[S₂[σ₂'1](σ+1)+1] - 1st σ=(next inaccessible+1)-stable
S[S₂[σ₂](S₂[σ₂'1](σ+1)+1)] - 1st σ=(next admissible after next inaccessible+1)-stable
S[S₂[σ₂'1](σ+2)] - 1st σ=(2nd next inaccessible)-stable
S[S₂[σ₂'1](S₂[σ₂'1](σ+1))] - 1st σ=(next next inaccessible)-stable
S[S₂[σ₂'1](1_|σ2|σ+1)] - 1st σ=(1st limit of next inaccessible)-stable
S[S₂[σ₂'1](1_|σ2'1|σ+1)] - 1st σ=(1st next 2-inaccessible)-stable
S[S₂[σ₂'1](1_|σ2'1|0_|σ2'1|σ+1)] - 1st σ=(1st next hyper-inaccessible)-stable
S[S₂[σ₂'1](1_|σ2'1|0_|σ2'1|0_|σ2'1|σ+1)] - 1st σ=((ϒ²)-order-inaccessible)-stable
S[S₂[σ₂'2](σ+1)] - 1st σ=(next Mahlo)-stable {zoo 2.12}
S[S₂[σ₂'2](σ+1)] - 1st σ=(next П₃-reflecting)-stable
S[S₂[σ₂'n](σ+1)] - 1st σ=(next П_n-reflecting)-stable
S[S₂[σ₂+1](σ+1)] - 1st σ=(next (+1)-stable)-stable; 1st doubly (+1)-stable; L_σ≺₁L_β≺₁L_β+1
S[S₂[σ₂+1](σ+1)+1] - 1st σ=(next (+1)-stable+2)-stable
S[S₂[σ₂+1](σ+1)+2] - 1st σ=(next (+1)-stable+3)-stable
S[S₂[σ₂+1](σ+1)+σ] - 1st σ=(next (+1)-stable+σ)-stable
S[S₂[σ₂+1](σ+2)] - 1st σ=(next 2nd (+1)-stable)-stable
S[S₂[σ₂+1](S[σ₂+1](σ+1))] - 1st σ=(next next (+1)-stable)-stable
S[S₂[σ₂+1](1_|σ2|σ+1)] - 1st σ=(1st limit of next (+1)-stable)-stable
S[S₂[σ₂+1](1_|σ2'1|σ+1)] - 1st σ=(1st next inaccessible limit of next (+1)-stable)-stable)
S[S₂[σ₂+1](1_{|σ2'1/σ2'2|}σ+1)] - 1st σ=(1st next Mahlo limit of next (+1)-stable)-stable)
S[S₂[σ₂+1](1_{|σ2'1/σ2+1|}σ+1)] - 1st σ=(1st next (+1)-stable limit of next (+1)-stable)-stable)
S[S₂[σ₂+1](1_|σ2'1|σ+1)] - 1st σ=(1st next Mahlo that is next (+1)-stable are stationary)-stable)
S[S₂[σ₂+1](1_|σ2+1|σ+1)] - 1st σ=(1st next 2-(+1)-stable are stationary)-stable)
S[S₂[σ₂+1'1](1_|σ2+1|σ+1)] - 1st σ=(1st next (+1)-П₁-reflecting)-stable)
S[S₂[σ₂+2](σ+1)] - 1st σ=(next (+2)-stable)-stable; 1st doubly (+2)-stable; L_σ≺₁L_β≺₁L_β+2 {zoo 2.13}
S[S₂[σ₂+α](σ+1)] - 1st σ=(next (+α)-stable)-stable; 1st doubly (+α)-stable
S[S₂[σ₂+σ](σ+1)] - 1st σ=(next (+σ)-stable)-stable;
S[S₂[σ₂+σ^σ](σ+1)] - 1st σ=(next (+σ^σ)-stable)-stable;
S[S₂[σ₂+S₂[σ₂](σ+1)](σ+1)] - 1st σ=(next (σ⁺)-stable)-stable;
S[S₂[σ₂+S₂[σ₂+1](σ+1)](σ+1)] - 1st σ=(next σ=(next (+1)-stable)-stable)-stable;
S[S₂[σ₂+S₂[σ₂+1](σ+σ)](σ+1)] - 1st σ=(next σ=(next (+σ)-stable)-stable)-stable;
S[S₂[σ₂+S₂[σ₂+S₂[σ₂+1](σ+σ)](σ+1)](σ+1)] - 1st σ=(next σ=(next σ=(next (+σ)-stable)-stable)-stable)-stable;  
S[S₂[σ₂+σ₂](σ+1)] - 1st σ=(next σ₂=(+σ₂)-stable)-stable; 1st doubly (+σ)-stable
S[S₂[ε_σ2+1](σ+1)] - 1st σ=(next σ₂=(+ε_σ2+1)-stable)-stable; 1st doubly (+ε_σ+1)-stable
S[S₂[S₃[σ₃](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next admissible)-stable)-stable
S[S₂[S₃[σ₃](σ₂+1)](σ+1)+1] - 1st σ=(next σ₂=(next admissible)-stable+2)-stable
S[S₂[S₃[σ₃](σ₂+1)](σ+1)+2] - 1st σ=(next σ₂=(next admissible)-stable+3)-stable
S[S₂[S₃[σ₃](σ₂+1)](σ+2)] - 1st σ=(next 2nd σ₂=(next admissible)-stable)-stable
S[S₂[S₃[σ₃](σ₂+1)+1](σ+1)] - 1st σ=(next σ₂=((next admissible)+2)-stable)-stable
S[S₂[S₃[σ₃](σ₂+2)](σ+1)] - 1st σ=(next σ₂=(next 2nd admissible)-stable)-stable
S[S₂[S₃[σ₃](σ₂+σ)](σ+1)] - 1st σ=(next σ₂=(next σ-th admissible)-stable)-stable
S[S₂[S₃[σ₃](σ₂+σ₂)](σ+1)] - 1st σ=(next σ₂=(next σ₂-th admissible)-stable)-stable
S[S₂[S₃[σ₃](S₃[σ₃](σ₂+1))](σ+1)] - 1st σ=(next σ₂=(next next admissible)-stable)-stable
S[S₂[S₃[σ₃](1_|σ|σ₂+1)](σ+1)] - 1st σ=(next σ₂=(limit of α-next admissible)-stable)-stable
S[S₂[S₃[σ₃](1_|σ2|σ₂+1)](σ+1)] - 1st σ=(next σ₂=(σ-next admissible)-stable)-stable
S[S₂[S₃[σ₃](1_|σ3|σ₂+1)](σ+1)] - 1st σ=(next σ₂=(σ₂-next admissible)-stable)-stable
S[S₂[S₃[σ₃'1](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next inaccessible)-stable)-stable
S[S₂[S₃[σ₃'2](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next Mahlo)-stable)-stable
S[S₂[S₃[σ₃'2](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next П₃-reflecting)-stable)-stable
S[S₂[S₃[σ₃'n](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next П_n-reflecting)-stable)-stable
S[S₂[S₃[σ₃+1](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next (+1)-stable)-stable)-stable; triply (+1)-stable; L_σ≺₁L_β≺₁L_γ≺₁L_γ+1
S[S₂[S₃[σ₃+1](σ₂+1)](σ+1)+1] - 1st σ=(next σ₂=(next (+1)-stable)-stable+2)-stable;
S[S₂[S₃[σ₃+1](σ₂+1)](σ+1)+σ] - 1st σ=(next σ₂=(next (+1)-stable)-stable+σ)-stable;
S[S₂[S₃[σ₃+1](σ₂+1)](σ+2)] - 1st σ=(next 2nd σ₂=(next (+1)-stable)-stable)-stable;
S[S₂[S₃[σ₃+1](σ₂+1)](σ+σ)] - 1st σ=(next σ-th σ₂=(next (+1)-stable)-stable)-stable;
S[S₂[S₃[σ₃+1](σ₂+1)+1](σ+1)] - 1st σ=(next σ₂=(next (+1)-stable+2)-stable)-stable;
S[S₂[S₃[σ₃+1](σ₂+1)+σ](σ+1)] - 1st σ=(next σ₂=(next (+1)-stable+σ)-stable)-stable;
S[S₂[S₃[σ₃+1](σ₂+1)+σ₂](σ+1)] - 1st σ=(next σ₂=(next (+1)-stable+σ₂)-stable)-stable;
S[S₂[S₃[σ₃+1](σ₂+2)](σ+1)] - 1st σ=(next σ₂=(next 2nd (+1)-stable)-stable)-stable;
S[S₂[S₃[σ₃+1](σ₂+σ)](σ+1)] - 1st σ=(next σ₂=(next σ-th (+1)-stable)-stable)-stable;
S[S₂[S₃[σ₃+1](σ₂+σ₂)](σ+1)] - 1st σ=(next σ₂=(next σ₂-th (+1)-stable)-stable)-stable;
S[S₂[S₃[σ₃+2](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next (+2)-stable)-stable)-stable
S[S₂[S₃[σ₃+α](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next (+α)-stable)-stable)-stable
S[S₂[S₃[σ₃+σ](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next (+σ)-stable)-stable)-stable
S[S₂[S₃[σ₃+σ₂](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next (+σ₂)-stable)-stable)-stable
S[S₂[S₃[σ₃+σ₃](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next σ₃=(+σ₃)-stable)-stable)-stable
S[S₂[S₃[S₄[σ₄+1](σ₃+1)](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next σ₃=(next (+1)-stable)-stable)-stable)-stable; quadruply (+1)-stable; L_σ≺₁L_β≺₁L_γ≺₁L_δ≺₁L_δ+1
S[...S_n[σ_n+1]...(σ+1)] - 1st n-ply (+1)-stable;
S[σ_ω] = S[S_ω[σ_ω]] = S[S_ω[σ_ω](σ+1)] = S[S₂[S_ω[σ_ω](σ+2)](σ+1)] = S[S₂[...S_n[...S_ω[σ_ω]...(σ_n+1)]...(σ+2)](σ+1)] - 1st ω-ply stable; 1st nonprojectable; 1st strongly Σ₁-admissible, [П¹₂-CA₀], [Δ¹₃-CA₀], {zoo 2.15}
S[σ_ω](1) - 2nd ω-ply stable
S[σ_ω](ω) - ω-th ω-ply stable
S[S_ω[σ_ω](σ+1)'1] - 1st (ω-ply stable)-П₁-reflecting; 1st admissible limit of ω-ply stable
S[S_ω[σ_ω](σ+1)+1] - 1st σ=(next σ₂=(...(ω-ply stable)...)-stable+2)-stable
S[S_ω[σ_ω](σ+1)+1](1) - 2nd σ=(next σ₂=(...(ω-ply (+1)-stable)...)-stable+2)-stable
S[S₂[S_ω[σ_ω](σ₂+1)+1](σ+1)] - 1st σ=(next σ₂=(next σ₃=(...(ω-ply (+1)-stable)...)-stable+2)-stable)-stable
S[S₂[S_ω[σ_ω](σ₂+2)](σ+1)] - 1st σ=(next σ₂=(2nd next σ₃=(...(ω-ply (+1)-stable)...)-stable)-stable)-stable
S[S₂[S₃[S_ω[σ_ω](σ₃+1)+1](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next σ₃=(next σ₄=(...(ω-ply (+1)-stable)...)-stable+2)-stable)-stable)-stable
S[S₂[S₃[S_ω[σ_ω](σ₃+2)](σ₂+1)](σ+1)] - 1st σ=(next σ₂=(next σ₃=(2nd next σ₄=(...(ω-ply (+1)-stable)...)-stable)-stable)-stable)-stable
S[S_ω[σ_ω](1)] - 1st (1st nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1)](1) - 2nd (1st nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](2)] - 1st (2nd nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](ω)] - 1st (ω-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](S[S_ω[σ_ω](ω)])] - 1st ((1st (ω-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](S[S_ω[σ_ω](S[S_ω[σ_ω](ω)])])] - 1st ((1st ((1st (ω-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)-stable
α↦S[S_ω[σ_ω](α)] = S[S_ω[σ_ω](α)] - 1st α↦((α-th nonprojectable limit of nonprojectables)-stable)
S[S_ω[σ_ω](σ)] - 1st σ=(σ-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](S₂[σ₂](σ+1)] - 1st σ=(σ⁺-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](S₂[S_ω[σ_ω](σ)])] - 1st σ=(((σ-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](S₂[S_ω[σ_ω](S₂[S_ω[σ_ω](σ)])])] - 1st σ=(((((σ-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)-stable
S[α↦S_ω[σ_ω](S₂[α])] - 1st α↦(σ=(α-th nonprojectable limit of nonprojectables)-stable)
S[S_ω[σ_ω](σ₂)] - 1st σ=(next σ₂=(...(σ₂-th nonprojectable limit of nonprojectables)...)-stable)-stable
S[S_ω[σ_ω](S₃[σ₃](σ₂+1)] - 1st σ=(next σ₂=(...(σ₂⁺-th nonprojectable limit of nonprojectables)...)-stable)-stable
S[S_ω[σ_ω](S₃[S_ω[σ_ω](σ₂)])] - 1st σ=(next σ₂=(...(((σ₂-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)...)-stable)-stable
S[S_ω[σ_ω](S₃[S_ω[σ_ω](S₃[S_ω[σ_ω](σ₂)])] - 1st σ=(next σ₂=(...(((((σ₂-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)-stable)-th nonprojectable limit of nonprojectables)...)-stable)-stable
S[α↦S_ω[σ_ω](S₃[α])] - 1st σ=(next α↦(σ₂=(...(α-th nonprojectable limit of nonprojectables)...)-stable))-stable
S[S_ω[σ_ω](σ₃)] - 1st σ=(next σ₂=(next σ₃=(...(σ₃-th nonprojectable limit of nonprojectables)...)-stable)-stable)-stable
S[S_ω[σ_ω](S[σ_ω])] - 1st (1st (1st nonprojectables)-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](S[σ_ω](1))] - 1st (1st (1st nonprojectables limit of nonprojectables)-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](S[σ_ω](S[σ_ω](1)))] - 1st (1st (1st (1st nonprojectables limit of nonprojectables)-th nonprojectables limit of nonprojectables)-th nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_|σ|0)] - 1st (1st fixed point of nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_|σ'1|0)] - 1st (1st inaccessible limit of  nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_|σ+1|0)] - 1st (1st (+1)-stable on nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_|σ+σ|0)] - 1st (1st σ=(+σ)-stable on nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_{|S2[σ2](σ+1)|}0)] - 1st (1st (next admissible)-stable on nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_{|S2[σ2+1](σ+1)|}0)] - 1st (1st doubly (+1)-stable on nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_{|S2[S3[σ3+1](σ2+1)](σ+1)|}0)] - 1st (1st triply (+1)-stable on nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_{|S2[Sω[σω]]|}0)] - 1st (1st nonprojectable on nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_{|S2[Sω[σω](1)]|}0)] - 1st (1st (1st nonprojectable limit of nonprojectables)-stable on nonprojectables)-stable
S[S_ω[σ_ω](1_{|α↦S2[Sω[σω](α)]|}0)] - 1st α↦(α-th nonprojectable on nonprojectable limit of nonprojectables)-stable
S[S_ω[σ_ω](1_|σ2|0)] - 1st σ=(next σ₂=(...(1st (σ-th nonprojectable limit of nonprojectables)-stable on nonprojectables)-stable...)-stable)-stable
S[S_ω[σ_ω](1_|σ3|0)] - 1st σ=(next σ₂=(next σ₃=(...(1st (σ₂-th nonprojectable limit of nonprojectables)-stable on nonprojectables)-stable...)-stable)-stable)-stable
S[S_ω[σ_ω](1_|σω|0)] - 1st (1st nonprojectable limit of nonprojectables on nonprojectable limit of nonprojectables)-stable = 1st (1st nonprojectable 2-limit of nonprojectables)-stable
S[S_ω[σ_ω](2_|σω|0)] - 1st (1st nonprojectable 3-limit of nonprojectables)-stable
S[S_ω[σ_ω](1_|σω|0_|σω|0)] - 1st (1st nonprojectable hyper-limit of nonprojectables)-stable
S[σ_ω'1] - 1st nonprojectable and admissible
S[σ_ω'1](1) - 2nd nonprojectable and admissible
S[S_ω[σ_ω'1](1)] - 1st nonprojectable and 2nd admissible
S[σ_ω'2] - 1st nonprojectable and Mahlo
S[σ_ω'2](1) - 2nd nonprojectable and Mahlo
S[S_ω[σ_ω'2](1)] - 1st nonprojectable and 2nd Mahlo
S[σ_ω'3] - 1st nonprojectable and П₃-reflecting
S[σ_ω'n] - 1st nonprojectable and П_n-reflecting
S[σ_ω+1] - 1st nonprojectable and (+1)-stable; 1st (ω+1)-ply (+1)-stable [KP+Σ₁-sep], [П¹₂-CA+BI]
S[S_ω[σ_ω+1]+1] - 1st σ=(next σ₂=(...((ω+1)-ply (+1)-stable)...)-stable+2)-stable
S[S₂[S_ω[σ_ω+1](σ+1)+1]] - 1st σ=(next σ₂=(next σ₃=(...((ω+1)-ply (+1)-stable)...)-stable+2)-stable)-stable
S[S₂[S₃[S_ω[σ_ω+1](σ₂+1)+1](σ+1)]] - 1st σ=(next σ₂=(next σ₃=(next σ₄=(...((ω+1)-ply (+1)-stable)...)-stable+2)-stable)-stable)-stable
S[S_ω[σ_ω+1](1)] - 1st nonprojectable limit of (ω+1)-ply (+1)-stable
S[σ_ω+1'1] - 1st (ω+1)-ply (+1)-stable and admissible
S[σ_ω+1'2] - 1st (ω+1)-ply (+1)-stable and Mahlo
S[σ_ω+1'n] - 1st (ω+1)-ply (+1)-stable and П_n-reflecting
S[σ_ω+2] - 1st nonprojectable and (+2)-stable; 1st (ω+1)-ply (+2)-stable
S[σ_ω+α] - 1st nonprojectable and (+α)-stable; 1st (ω+1)-ply (+α)-stable
S[σ_ω+σ] - 1st σ=(ω+1)-ply (+σ)-stable
S[σ_ω+σ+1] - 1st σ=(ω+1)-ply (+σ+1)-stable
S[σ_ω+S₂[σ₂](σ+1)] - 1st σ=(ω+1)-ply (σ⁺)-stable
S[σ_ω+S₂[σ₂+1](σ+1)] - 1st σ=(ω+1)-ply (next (+1)-stable)-stable
S[σ_ω+S₂[S₃[σ₃](σ₂+1)](σ+1)] - 1st σ=(ω+1)-ply (σ⁺)-stable (next σ₂=(next (+1)-stable)-stable)-stable
S[σ_ω+S₂[S_ω[σ_ω](σ+1)]] - 1st σ=(ω+1)-ply σ=(ω-ply stable)-stable
S[σ_ω+σ₂] - 1st σ=(next σ₂=(next (...(ω+1)-ply (+σ₂)-stable)...)-stable)-stable)-stable
S[σ_ω+S₃[σ₃](σ₂+1)] - 1st σ=(next σ₂=(next (...(ω+1)-ply (+σ₂⁺)-stable)...)-stable)-stable)-stable
S[σ_ω+S₃[S_ω[σ_ω](σ₂+1)]] - 1st σ=(next σ₂=(next (...(ω+1)-ply σ₂=(ω-ply stable)-stable)...)-stable)-stable)-stable
S[σ_ω+σ₃] - 1st σ=(next σ₂=(next σ₃=(...(ω+1)-ply (+σ₃)-stable)...)-stable)-stable)-stable
S[σ_ω+S₄[σ₄](σ₃+1)] - 1st σ=(next σ₂=(next σ₃=(...(ω+1)-ply (+σ₃⁺)-stable)...)-stable)-stable)-stable
S[σ_ω+S₄[S_ω[σ_ω](σ₃+1)]] - 1st σ=(next σ₂=(next σ₃=(...(ω+1)-ply σ₃=(ω-ply stable)-stable)...)-stable)-stable)-stable
S[σ_ω+σ_ω] - 1st nonprojectable and σ=(+σ)-stable; 1st (ω+1)-ply σ=(+σ)-stable
S[σ_ω+σ_ω+1] - 1st nonprojectable and σ=(+σ+1)-stable; 1st (ω+1)-ply σ=(+σ+1)-stable
S[S_ω+1[σ_ω+1](σ_ω+1)] - 1st nonprojectable and σ=(σ⁺)-stable; 1st (ω+1)-ply σ=(σ⁺)-stable
S[S_ω+1[σ_ω+1](σ_ω+2)] - 1st nonprojectable and σ=(σ⁺⁺)-stable; 1st (ω+1)-ply σ=(σ⁺⁺)-stable
S[S_ω+1[σ_ω+1](S[σ_ω+1](σ_ω+1))] - 1st nonprojectable and σ=(next next admissible)-stable; 1st (ω+1)-ply σ=(next next admissible)-stable
S[S_ω+1[σ_ω+1'1](σ_ω+1)] - 1st nonprojectable and (next inaccessible)-stable; 1st (ω+1)-ply (next inaccessible)-stable
S[S_ω+1[σ_ω+1'2](σ_ω+1)] - 1st nonprojectable and (next Mahlo)-stable; 1st (ω+1)-ply (next Mahlo)-stable
S[S_ω+1[σ_ω+1+1](σ_ω+1)] - 1st nonprojectable and doubly (+1)-stable; 1st (ω+2)-ply (+1)-stable
S[S_ω+1[S_ω+2[σ_ω+2](σ_ω+1+1)](σ_ω+1)] - 1st nonprojectable and doubly (next admissible)-stable; 1st (ω+2)-ply (next admissible)-stable
S[S_ω+1[S_ω+2[σ_ω+2'1](σ_ω+1+1)](σ_ω+1)] - 1st nonprojectable and doubly (next inaccessible)-stable; 1st (ω+2)-ply (next inaccessible)-stable
S[S_ω+1[S_ω+2[σ_ω+2'2](σ_ω+1+1)](σ_ω+1)] - 1st nonprojectable and doubly (next Mahlo)-stable; 1st (ω+2)-ply (next Mahlo)-stable
S[S_ω+1[S_ω+2[σ_ω+2+1](σ_ω+1+1)](σ_ω+1)] - 1st nonprojectable and triply (+1)-stable; 1st (ω+3)-ply (+1)-stable
S[S_ω+1[S_ω+2[S_ω+3[σ_ω+3+1](σ_ω+2+1)](σ_ω+1+1)](σ_ω+1)] - 1st nonprojectable and quadruply (+1)-stable; 1st (ω+4)-ply (+1)-stable
S[σ_ω×2] - 1st doubly nonprojectable; 1st (ω×2)-ply stable
S[σ_ω×2+1] - 1st doubly nonprojectable and (+1)-stable; 1st (ω×2+1)-ply stable
S[S_ω×2+1[σ_ω×2+1+1](σ_ω×2+1)] - 1st doubly nonprojectable and doubly (+1)-stable; 1st (ω×2+2)-ply stable
S[σ_ω×3] - 1st triply nonprojectable; 1st (ω×3)-ply stable
S[σ_ε0] - 1st ε₀-ply stable [Δ¹₃-CA]
S[σ_α] = α↦S[σ_α] - 1st α-ply stable
S[σ_σ] - σ=σ-ply stable
S[σ_σ+1] - σ=σ+1-ply stable
S[σ_εσ+1] - σ=ε_σ+1-ply stable
S[σ_{S2[σ2](σ+1)}] - σ=(σ⁺)-ply stable
S[σ_{S2[σ2+1](σ+1)}] - σ=(next (+1)-stable)-ply stable
S[σ_S[σω]] - σ=(next nonprojectable)-ply stable
S[σ_S[σσ]] - σ=((σ-ply stable)-ply stable)
S[σ_{S[σS2[σ2](σ+1)]}] - σ=((next (+1)-stable)-ply stable)-ply stable
S[σ_S[σS[σσ]]] - σ=(((σ-ply stable)-ply stable)-ply stable
S[σ_S[α]] = S[α↦S[σ_α]] - σ=(1st α-ply stable)
S[σ_σ2] - σ=(σ₂=(σ₂-ply stable)-ply stable); doubly-ply stable
S[σ_{S[σ3](σ2+1)}] - σ=(σ₂=(σ₂⁺-ply stable)-ply stable)-ply stable
S[σ_σ3] = σ=(σ₂=(σ₃=(σ₃-ply stable)-ply stable)-ply stable; triply-ply stable
S[σ_σn] = σ=(σ₂=(σ₃=(...=(σ_n-ply stable)...-ply stable)-ply stable)-ply stable; σ_n-ply stable; n-ply-ply stable
S[σ_σω] = σ_ω-ply stable; ω-ply-ply stable
S[S_σω[σ_σω]+1] = 1st σ=(next (...(σ_ω-ply stable)...)-stable+2)-stable
S[S₂[S_σω[σ_σω](σ+1)+1]] = 1st σ=(next (...(σ_ω-ply stable)...)-stable+2)-stable
S[S₂[S₃[S_σω[σ_σω](σ₂+1)+1](σ+1)]] - 1st σ=(next σ₂=(next σ₃=(...(σ_ω-ply stable)...)-stable+2)-stable)-stable
S[S_ω[S_σω[σ_σω](σ_ω+1)+1]] = 1st σ=(...(ω-ply ...(σ_ω-ply stable)...)-stable+2)...)-stable
S[S_σ[S_σω[σ_σω](σ_σ+1)+1]] = 1st σ=(σ₂=(σ₃=(...=(σ_ω-ply stable)...-ply stable)-ply stable+2)-ply stable
S[S_σ2[S_σω[σ_σω](σ_σ2+1)+1]] = 1st σ=(σ₂=(σ₃=(...=(σ_ω-ply stable)...-ply stable+2)-ply stable)-ply stable
S[S_σω[σ_σω](1)] - 1st (1st σ_ω-ply stable limit of  σ_ω-ply stables)-stable
S[S_σω[σ_σω](2)] - 1st (2nd σ_ω-ply stable limit of  σ_ω-ply stables)-stable
S[S_σω[σ_σω](σ)] - 1st σ=(σ-th σ_ω-ply stable limit of  σ_ω-ply stables)-stable
S[S_σω[σ_σω](σ₂)] - 1st σ=(next σ₂=(...(σ₂-th σ_ω-ply stable limit of  σ_ω-ply stables)...)-stable)-stable
S[S_σω[σ_σω](1_|σ|0)] - 1st (1st fixed point of σ_ω-ply stable limit of  σ_ω-ply stables)-stable
S[S_σω[σ_σω](1_|σ2|0)] - 1st σ=(next σ₂=(...(1st (σ-th σ_ω-ply stable limit of σ_ω-ply stable)-stable on σ_ω-ply stable)-stable...)-stable)-stable
S[S_σω[σ_σω](1_|σω|0)] - 1st (1st nonprojectable limit of nonprojectables on σ_ω-ply stable limit of σ_ω-ply stable)-stable
S[S_σω[σ_σω](1_|σσ|0)] - 1st σ=((σ-th σ_ω-ply stable limit of σ_ω-ply stable)-stable)-ply stable
S[S_σω[σ_σω](1_|σσ2|0)] - 1st σ=(σ₂=((σ₂-th σ_ω-ply stable limit of σ_ω-ply stable)-ply stable)-ply stable
S[S_σω[σ_σω](1_|σσω|0)] - 1st (1st σ_ω-ply stable limit of σ_ω-ply stable on σ_ω-ply stable limit of σ_ω-ply stable)-stable = 1st (1st σ_ω-ply stable 2-limit of  σ_ω-ply stables)-stable
S[S_σω[σ_σω](1_|σσω|0_|σσω|0)] - 1st (1st σ_ω-ply stable hyper-limit of  σ_ω-ply stables)-stable
S[σ_σω'1] - 1st σ_ω-ply stable and admissible
S[σ_σω'2] - 1st σ_ω-ply stable and Mahlo
S[σ_σω+1] - 1st σ_ω-ply stable and (+1)-stable
S[σ_σω+σ] - 1st σ=σ_ω-ply stable and (+σ)-stable
S[σ_σω+σ₂] - 1st σ=(next σ₂=(next (...(σ_ω-ply stable and (+σ)-stable)...)-stable)-stable)-stable
S[σ_σω+σ_σ] - 1st σ=(σ_ω-ply stable and (+σ)-stable)-ply stable
S[σ_σω+σ_σ2] - 1st σ=(σ₂=(σ_ω-ply stable and (+σ₂)-stable)-ply stable
S[σ_σω+σ_σω] - 1st σ_ω-ply stable and σ=(+σ)-stable
S[σ_σω+1] - 1st σ_ω-ply stable and doubly stable = 1st (σ_ω+1)-ply stable and doubly stable
S[σ_σω+ω] - 1st σ_ω-ply stable and nonprojectable
S[σ_σω+σ] - 1st σ=σ_ω-ply stable and σ-ply stable
S[σ_σω+σ2] - 1st σ=(σ₂=...(σ_ω-ply stable and σ₂-ply stable)...-ply stable)-ply stable
S[σ_σω+σ3] - 1st σ=(σ₂=(σ₃=...(σ_ω-ply stable and σ₃-ply stable)...-ply stable)-ply stable)-ply stable
S[σ_σω+σω] - 1st σ_ω-ply stable and σ=(σ-ply stable)
S[σ_σω+1] - 1st σ_ω+1-ply stable
S[σ_σσ] = 1st σ=σ_σ-ply stable; σ=(σ-ply-ply stable)
S[σ_σσ2] = doubly-ply-ply stable
S[σ_σσn] = n-ply-ply-ply stable
S[σ_σσn] = n-ply-ply-ply stable
S[α↦σ_α] = S[σ_σσ...] = 1st (1st limit of ...-ply stable) [П¹₂-TR₀]

Up to Z₂:
Add a little inductivity to notation. Let:
S[σ] = S[SS[σσ]]
S[σ₂] = S[SS[σσ](1)]
S[σ_ω] = S[SS[σσ](ω)]
S[σ_σ] = S[SS[σσ](SS[σσ])]
S[σ_σσ] = S[SS[σσ](SS[σσ](SS[σσ])))]
S[σ_σσ] = S[SS[σσ](SS[σσ](SS[σσ])))]
S[α↦σ_α] = S[α↦SS[σσ](α)] = S[SS[σσ](1_|σσ|0)]]
Then we get the following extension:

S[SS[σσ](1_|σσ|0)] = S[SS[σσ](SS[σσ](SS[σσ](...))))] - 1st (1st limit of ...-ply stable) [П¹₂-TR₀]
S[SS[σσ](1_|σσ|1)] - 1st (2nd limit of ...-ply stable)
S[SS[σσ](2_|σσ|0)] - 1st (fixed point of limit of ...-ply stable); 1st (1st 2-limit of ...-ply stable)
S[SS[σσ](1_|σσ|0_|σσ|0)] - hyper-limit of ...-ply stable
S[SS[σσ'1]] - 1st σ=П₂-(St)-reflecting, where St - set of stable below; σ=П₂-reflecting on (β<σ|L_β≺₁L_σ); σ in which ...-ply stable are stationary [KP+Δ₂-sep], [Δ¹₃-CA+BI], {zoo 2.16}
S[SS[σσ'1]](1) - 2nd σ=П₂-(St)-reflecting
S[SS[σσ'1]](ω) - 1st limit of σ=П₂-(St)-reflecting
S[SS[σσ'1]'1] - 1st admissible limit of σ=П₂-(St)-reflecting; П₁-reflecting on σ=П₂-(St)-reflecting
S[SS[σσ'1]'2] - П₂-reflecting on σ=П₂-(St)-reflecting
S[SS[σσ'1]'3] - П₃-reflecting on σ=П₂-(St)-reflecting
S[SS[σσ'1]+1] = S[S_SS[σσ'1][SS[σσ'1]](σ+1)+1] - 1st (next (...(σ=П₂-(St)-reflecting)...)-stable+2)-stable
S[S₂[S_SS[σσ'1][SS[σσ'1]](σ+1)+1]] - 1st (next (next (...(σ=П₂-(St)-reflecting)...)-stable+2)-stable)-stable
S[S₂[S₃[S_SS[σσ'1][SS[σσ'1]](σ₃+1)+1](σ₂+1)](σ+1)] - 1st (next (next (next (...(σ=П₂-(St)-reflecting)...)-stable+2)-stable)-stable)-stable
S[S_ω[S_SS[σσ'1][SS[σσ'1]](σ_ω)](1)] - 1st (1st nonprojectable limit of nonprojectables on σ=П₂-(St)-reflecting)-stable
S[S_σ[S_SS[σσ'1][SS[σσ'1]](σ_σ)](1)] - 1st (1st σ=(σ-ply stable) limit of σ=(σ-ply stable) on σ=П₂-(St)-reflecting)-stable
S[S_σσ[S_SS[σσ'1][SS[σσ'1]](σ_σσ)](1)] - 1st (1st σ=(σ_σ-ply stable) limit of σ=(σ_σ-ply stable) on σ=П₂-(St)-reflecting)-stable
S[S_SS[σσ'1][SS[σσ'1]](1)] - 1st (1st σ=П₂-(St)-reflecting limit of σ=П₂-(St)-reflecting)-stable
S[S_SS[σσ'1][SS[σσ'1]'1]] -  1st (1st σ=П₂-(St)-reflecting admissible limit of σ=П₂-(St)-reflecting)-stable
S[S_SS[σσ'1][SS[σσ'1]'2]] -  1st (1st σ=П₂-(St)-reflecting Mahlo limit of σ=П₂-(St)-reflecting)-stable
S[S_SS[σσ'1][SS[σσ'1]'3]] -  1st σ=П₂-(St)-reflecting and П₃-reflecting
S[S_SS[σσ'1][SS[σσ'1]+1]] - 1st σ=П₂-(St)-reflecting and (+1)-stable
S[S_SS[σσ'1][ε_SS[σσ'1]+1]] - 1st σ=П₂-(St)-reflecting and (+ε_σ+1)-stable
S[S_{SS[σσ](SS[σσ'1]+1)}[SS[σσ](SS[σσ'1]+1)]SS[σσ'1]+1)] - 1st σ=П₂-(St)-reflecting and (σ⁺)-stable
S[SS[σσ](SS[σσ'1]+ω)] - 1st σ=П₂-(St)-reflecting and ω-ply stable
S[SS[σσ](SS[σσ'1]+SS[σσ'1])] - 1st σ=П₂-(St)-reflecting and σ-ply stable
S[SS[σσ](SS[σσ](SS[σσ'1]+ω))] - 1st σ=П₂-(St)-reflecting and σ_ω-ply stable
S[SS[σσ](SS[σσ](SS[σσ'1]+SS[σσ'1]))] - 1st σ=П₂-(St)-reflecting and σ_σ-ply stable
S[SS[σσ](1_|σσ|SS[σσ'1]+1)] - 1st σ=П₂-(St)-reflecting and σ_σ-ply stable
S[SS[σσ'1](1)] - 1st σ=2nd П₂-(St)-reflecting
S[SS[σσ'1](ω)] - 1st σ=1st limit of П₂-(St)-reflecting
S[SS[σσ'1](1_|σσ|0)] - 1st σ=1s fixed point limit of П₂-(St)-reflecting
S[SS[σσ'1](1_|σσ'1|0)] - 1st σ=1st admissible limit of П₂-(St)-reflecting; 1st σ=П₁-(St)-reflecting on П₂-(St)-reflecting
S[SS[σσ'2]] - 1st σ=П₂-(St)-reflecting on П₂-(St)-reflecting;
S[SS[σσ'3]] - 1st σ=П₃-(St)-reflecting; σ=П₃-reflecting on (β<σ|L_β≺₁L_σ);
S[SS[σσ'n]] - 1st σ=П_n-(St)-reflecting; σ=П_n-reflecting on (β<σ|L_β≺₁L_σ);
S[SS[σσ+1]] - 1st (+1)-2-stable; L_σσ≺₂L_St+1; σ=(+1)-stable on (β<σ|L_β≺₁L_σ)
S[SS[σσ+1]](1) -2nd (1st (+1)-2-stable)-stable
S[SS[σσ+1](1)] - 1st (2nd (+1)-2-stable)-stable
S[SS[σσ+1'1]] - 1st (+1)-П₁-(St)-reflecting; 1st ((1st admissible after ϒ)-order-(+1)-2-stable)-stable
S[SS[σσ+1'2]] - 1st (+1)-П₂-(St)-reflecting
S[SS[σσ+2]] - 1st (+2)-2-stable; L_σσ≺₂L_St+2; σ=(+2)-stable on (β<σ|L_β≺₁L_σ)
S[SS[σσ+σ]] - 1st σ=(1st (+σ)-2-stable)-stable
S[SS[σσ+σ₂]] - 1st σ=(next σ₂=(next...(1st (+σ₂)-2-stable)...-stable)-stable)-stable
S[SS[σσ+σσ]] - 1st σ=(+σ)-2-stable; L_σσ≺₂L_St+σσ; σ=(+σ)-stable on (β<σ|L_β≺₁L_σ)
S[SS[S[σ](σσ+1)]] - 1st (next admissible)-2-stable
S[SS[S[σ+1](σσ+1)]] - 1st (next (+1)-stable)-2-stable
S[SS[S[σ_ω](σσ+1)]] - 1st (next nonprojectable)-2-stable
S[SS[S[σ_σ](σσ+1)]] - 1st (next σ=(σ-ply stable))-2-stable
S[SS[S[SS₂[σσ₂'1]](σσ+1)]] - 1st (next П₂-(St)-reflecting)-2-stable
S[SS[S[SS₂[σσ₂+1]](σσ+1)]] - 1st (next 2-stable)-2-stable = doubly (+1)-2-stable; L_σσ≺₂ L_β≺₂L_St+1
S[SS[S[SS₂[S[SS₃[σσ₃'1]](σσ₂+1)]](σσ+1)]] - 1st (next (next 2-stable)-2-stable)-2-stable = triply (+1)-2-stable; L_σσ≺₂ L_β≺ L_γ≺₂L_St+1
S[SS_ω[σσ_ω]] - 1st ω-ply 2-stable; 2-nonprojectable; strongly Σ₂-admissible [П¹₃-CA₀], [Δ¹₄-CA₀]
S[SS_ω[σσ_ω+1]] - 1st (ω+1)-ply 2-stable [KP+Σ₂-sep], [П¹₃-CA+BI]
S[SS_ω[S[SS_ω+1[σσ_ω+1+1]](σσ+1)]] - 1st (ω+2)-ply 2-stable
S[SS[σσ_ε0]] - 1st (ε₀)-ply 2-stable [Δ¹₄-CA]
S[SS[σσ_σ]] - 1st σ=(1st σ-ply 2-stable)-stable
S[SS[σσ_σσ]] - 1st σ=(σ-ply 2-stable)
S[SS[σσ_σσσσ]] = 1st σ=(σ-ply-ply 2-stable)
S[SS[SSS[σσσ](1_|σσσ|0)]]] - 1st (1st limit of ...-ply 2-stable) [П¹₃-TR₀]
S[SS[SSS[σσσ'1]]] - П₂-(St₂)-reflecting, where St₂ - set of 2-stable below; inaccessible limit of ...-ply 2-stable, [KP+Δ₃-sep], [Δ¹₄-CA+BI]
S[SS[SSS[σσσ+1]]] - (+1)-3-stable; L_σσ≺₃L_St2+1
S[SS[SSS_ω[σσσ_ω]]]- ω-ply 3-stable; 3-nonprojectable; strongly Σ₃-admissible [П¹₄-CA₀], [Δ¹₅-CA₀]
S[SS[SSS_ω[σσσ_ω+1]]] - (ω+1)-ply 3-stable [KP+Σ₃-sep], [П¹₃-CA+BI]
S[SS[SSS_ε0[σσσ_ε0]]] - (ε₀)-ply 3-stable [Δ¹₅-CA]
S[SS[SSS[SSSS[σσσσ'1]]]] - П₂-(St₃)-reflecting, where St₃ - set of 3-stable below; inaccessible limit of ...-ply 3-stable, [KP+Δ₄-sep], [Δ¹₅-CA+BI]
S[SS[SSS[SSSS_ω[σσσσ_ω]]]] - ω-ply 4-stable; 4-nonprojectable; strongly Σ₄-admissible [П¹₅-CA₀], [Δ¹₆-CA₀]
S[SS[SSS[SSSS_ω[σσσσ_ω+1]]]] - (ω+1)-ply 4-stable [KP+Σ₄-sep], [П¹₄-CA+BI]
S[SS[SSS[SSSS_ε0[σσσσ_ε0]]]] - (ε₀)-ply 4-stable [Δ¹₆-CA]
S[SS[SSS[SSSS[SSSSS[σσσσσ'1]]]]] - П₂-(St₄)-reflecting, where St₄ - set of 4-stable below; inaccessible limit of ...-ply 4-stable, [KP+Δ₅-sep], [Δ¹₆-CA+BI]

Up to ZFC:
Next comes a very speculative guess.
Let's add more inductivity to notation.
S[σ] = S⁽¹⁾[σ⁽¹⁾] = S_G[g][G[g]]
S₂[σ₂] = S₂⁽¹⁾[σ₂⁽¹⁾] = S_G[g](1)[G[g](1)]
SS[σσ] = S⁽²⁾[σ⁽²⁾] = S_G[g'1][G[g'1]]
SS₂[σσ₂] = S₂⁽²⁾[σ₂⁽²⁾] = S_G[g'1](1)[G[g'1](1)]
SSS[σσσ] = S⁽³⁾[σ⁽³⁾] = S_G[g'2][G[g'2]]
SSS₃[σσσ₂] = S₂⁽³⁾[σ₂⁽³⁾] = S_G[g'2](1)[G[g'2](1)]
e.t.c.
Then we get the following extension:

S[S^(ω)[σ^(ω)]] = S[S_G[g+1][G[g+1]]] = G[g+1] - start 1st 2nd-order gap length 1 and (+1)-stable; (L_β/L_β+1)∩ω₁=∅ [Z₂], [ZFC^-], {zoo 2.17}
S[S^(ω)[σ^(ω)]+1] = S[S_G[g+1][G[g+1]]+1] - start 1st 2nd-order gap length 1 and (+2)-stable
S[S^(ω+1)[σ^(ω+1)]] = S[S_G[g+1'1][G[g+1'1]]] - start 1st 2nd-order gap length 1 and (+1)-2-stable
S[S^(ω+1)[σ^(ω+1)]+1] = S[S_G[g+1'1][G[g+1'1]]+1] - start 1st 2nd-order gap length 1 and (+2)-2-stable
S[S^(ω×2)[σ^(ω×2)]] = S[S_G[g+2][G[g+2]]] - start 1st 2nd-order gap length 2 and (+1)-stable; (L_β/L_β+2)∩ω₁=∅ {zoo 2.18}
S[S^(ω2)[σ^(ω2)]] = S[S_G[g+ω][G[g+ω]]] - start 1st 2nd-order gap length ω and (+1)-stable; (L_β/L_β+ω)∩ω₁=∅
S[S^(ε₀)[σ^(ε₀)]] = S[S_G[g+ε0][G[g+ε₀]]] - start 1st 2nd-order gap length ε₀ and (+1)-stable; (L_β/L_β+ε0)∩ω₁=∅
S[S^(α)[σ^(α)]] = S[S_G[g+α][G[g+α]]] - start 1st 2nd-order gap length α; (L_β/L_β+α)∩ω₁=∅
S[S_G[g×2][G[g×2]]] = G[g×2] - β=(start 1st 2nd-order gap length β); (L_β/L_β+β)∩ω₁=∅ {zoo 2.19}
S[S_G[g^g][G[g^g]]] = G[g^g] - β=(start 1st 2nd-order gap length β^β) {M. Srebrny 1973, Corollary 4.12.}
S[S_G[εg+1][G[ε_g+1]]] = G[ε_g+1] - β=(start 1st 2nd-order gap length β^β)
G[S[σ](g+1)] - β=(start 1st 2nd-order gap length next admissible after β) [KP+"ω₁ exists"], {zoo 2.21}
G[S[σ](g+ω)] - β=(start 1st 2nd-order gap length next ω-th admissible after β) [П²₁-CA₀], [Δ²₂-CA₀]
G[S[σ+1](g+1)] - β=(start 1st 2nd-order gap length next (+1)-stable after β)
G[S[S_ω[σ_ω]](g+1)] - β=(start 1st 2nd-order gap length ω-ple stable after β) [П²₂-CA₀], [Δ²₃-CA₀]
G[S[SS[σσ+1]](g+1)] - β=(start 1st 2nd-order gap length next (+1)-2-stable after β)
G[S[SS_ω[σσ_ω]](g+1)] - β=(start 1st 2nd-order gap length ω-ple 2-stable after β) [П²₃-CA₀], [Δ²₄-CA₀]
G[S[SS[SSS[σσσ+1]]](g+1)] - β=(start 1st 2nd-order gap length next (+1)-3-stable after β)
G[S[SS[SSS_ω[σσσ_ω]]](g+1)] - β=(start 1st 2nd-order gap length ω-ple 2-stable after β) [П²₄-CA₀], [Δ²₅-CA₀]
G[G₂[g₂+1](g+1)] - start 1st 3d-order gap length 1; (L_β/L_β+1)∩ω₂=∅ [Z₃], [ZFC^-+"ω₁ exists"], {zoo 2.22}
G[G₂[S[σ](g₂+1)](g+1)] - β=(start 1st 3d-order gap length next admissible after β) [KP+"ω₂ exists"]
G[G₂[S[σ](g₂+ω)](g+1)] - β=(start 1st 3d-order gap length next ω-th admissible after β) [П³₁-CA₀], [Δ³₂-CA₀]
G[G₂[S[S_ω[σ_ω]](g₂+1)](g+1)] - β=(start 1st 3d-order gap length ω-ple stable after β) [П³₂-CA₀], [Δ³₃-CA₀]
G[G₂[G₃[g₃+1](g₂+1)](g+1)] - start 1st 4th-order gap length 1; (L_β/L_β+1)∩ω₃=∅ [Z₄], [ZFC^-+"ω₂ exists"]
G[G₂[G₃[S[σ](g₃+1)](g₂+1)](g+1)] - β=(start 1st 4th-order gap length next admissible after β) [KP+"ω₃ exists"]
G[G₂[G₃[S[σ](g₃+ω)](g₂+1)](g+1)] - β=(start 1st 4th-order gap length next ω-th admissible after β) [П⁴₁-CA₀], [Δ⁴₂-CA₀]
G[G₂[G₃[S[S_ω[σ_ω]](g₃+1)](g₂+1)](g+1)] - β=(start 1st 4th-order gap length ω-ple stable after β) [П⁴₂-CA₀], [Δ⁴₃-CA₀]
G[G₂[G₃[G₄[g₄+1](g₂+1)](g₂+1)](g+1)] - start 1st 5th-order gap length 1; (L_β/L_β+1)∩ω₄=∅ [Z₅], [ZFC^-+"ω₃ exists"]
G[...G_n[g_n+1]...(g+1)] - start 1st n-th-order gap length 1; (L_β/L_β+1)∩ω_n=∅ [Z_n], [ZFC^-+"ω_n exists"]
G[g_ω] - [ZFC^-+"ω_ω exists"], [Limit of HOA], [loader.c]
G[g_g] - [ZFC^-+"ω_(n<ω1) exists"]
G[g_g2] - [ZFC^-+"ω_ω1 exists"]
G[g_gω] - [ZFC^-+"ω_ωω exists"]
G[g_gg] - [ZFC^-+"ω_ω(n<ω1) exists"]
G[g_gg2] - [ZFC^-+"ω_ωω1 exists"]
G[GG[gg](1_|gg|0)] - [ZFC^-+"1st fixed point of ω_α exists"]
G[GG[gg'1]] - [ZFC^-+"predicate-free П¹₀-indescribable exists"]
G[GG[gg'2]] - [ZFC^-+"predicate-free П¹₀-indescribable on predicate-free П¹₀-indescribable exists"]
G[GG[gg'3]] - [ZFC^-+"predicate-free П¹₁-indescribable exists"]
G[GG[gg+1]] - [ZFC^-+"Σ₁-extendible exists"]
G[GG[GGG[ggg+1]]] - [ZFC^-+"Σ₂-extendible exists"]
G[GG[GGG[GGGG[gggg+1]]]] - [ZFC^-+"Σ₃-extendible exists"]
G[GG[GGG[GGGG[GGGGG[gggg+1]]]]] - [ZFC^-+"Σ₄-extendible exists"]
G[G^(ω)[g^(ω)]] - [ZFC], [ZFC^-+"worldly cardinal exists"], {zoo 2.24}

And my last guess. If we could repeat this iteration again but at the highest level, we would get:
S[S^(ω)[σ^(ω)]] = G[g+1] - [Z₂]
G[G^(ω)[g^(ω)]] = M[m+1] - [ZFC]
M[M^(ω)[m^(ω)]] - [MK], [ZFC^-+"inaccessible cardinal exists"] {M. Srebrny 1973, §7}
Then you can continue inductively extend notation
S = $1$; σ = &1&
G = $2$; g = &2&
M = $3$; m = &3&
e.t.c., then
$1$[&1&+1] - [KP+П_ω-ref]
$2$[&2&+1] - [Z₂], [ZFC^-]
$3$[&3&+1] - [ZFC], [ZFC^-+worldly cardinal]
$4$[&4&+1] - [2nd order extention of ZFC], [ZFC^-+"inaccessible cardinal exists"]
$5$[&5&+1] - [3d order extention of ZFC], [ZFC^-+"two inaccessible cardinal exists"]
$n$[&n&+1] - [n-th order extention of ZFC], [ZFC^-+"n inaccessible cardinal exists"]
Further I do not dare to make assumptions. Further theories are strengthened by the introduction of large cardinals. We all know how similar are the hierarchies of large cardinals and hierarchies of large countable ordinals. So I suppose that such recursive transformations continue to occur only at a higher level.
Inaccessible is cardinal analog of recursively inaccessible ordinal.
Mahlo is cardinal analog of recursively mahlo ordinal.
П¹_n-indescribable is cardinal analog of П_n+2-reflecting ordinal;
α-П¹_n-indescribable is cardinal analog of (+α)-П_n-reflecting ordinal;
Subtle is cardinal analog of 1st σ=(f(σ))-stable ordinal, where f() - some Δ₁-function;
n-ineffable is cardinal analog of Σ_n-stable ordinal, where n > 1;
α-Erdos is cardinal analog of gap ordinals;
Ramsey is cardinal analog of ordinals like "model of ZFC^-+predicate-free П¹₀-indescribable exists" and large
Actually, сardinal structure beatwen Subtle cardinal and Ramsey cardinal is poorly understood, but  Taranovsky attempted to describe it in terms of Reflective Cardinals. 
Then comes the new conceptual race. Measurable cardinals. The transition to it can be compared with the transition from countable to uncountable. This gives the potential for a new round of recursions. And Woodin cardinal is stationary set of measurable cardinals - limit of this recursive schema. As it is known, Z₂+PD or ZFC^-+"n Woodin cardinal exists" is the limit of TON.