User blog:Scorcher007/Large countable ordinal notation up to Z2 and ZFC

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[here outdated and incorrect information]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:

&upsih; - means pseudo-ordinal term used as diagonalizer for thing like &quot;hyper- x &quot;

(...0undefined0undefined0undefinedn) = (n)

(nundefined0undefined0undefined0undefined... m -times) = (nundefined)

k - meas property of ordinal

0 (&sigma;) - order

1 (&sigma;&#39;1) - inaccessibility

2 (&sigma;&#39;2) - mahloness

3 (&sigma;&#39;3) - П3-reflecting

e.t.c

thing like 1/2&#123;a&#125;, 1/2&#123;a&#125;/3&#123;b&#125;&#123;c&#123;d&#125;&#125;&#123;&#123;e&#125;&#125; means combination of this property

zoo - means refer to Madore D., Zoo of ordinals, 2017.

Up to П12-CA0:

S[&sigma;] - &omega;; 1st admissible  &#123;zoo 1.4&#125;

S[&sigma;](1) - 1st admissible after &omega;; &omega;1CK; &Delta;11-set of &omega; undefined [KP&omega;], &#123;zoo 2.1&#125;, collapse&#123;zoo 1.20&#125;

S[&sigma;](2) - 2nd admissible after &omega;; &omega;2CK; &Delta;11-set of &omega;1CK

S[&sigma;](2) - n-th admissible [П11-CA0], [&Delta;12-CA0]

S[&sigma;](&omega;) - 1st limit of admissible [KPl], [П11-CA+BI], &#123;zoo 2.2&#125; collapse&#123;zoo 1.21&#125;

S[&sigma;](&epsilon;0) - &epsilon;0-th admissible [&Delta;12-CA]

S[&sigma;](S[&sigma;](1)) - (1st admissible)-th admissible

S[&sigma;](1undefined0) - 1st fixed point of admissible

S[&sigma;](1undefined1) - 2nd fixed point of admissible

S[&sigma;](2undefined0) - 1st fixed point of fixed point of admissible = 1st 2-fixed point of admissible

S[&sigma;](1undefined0undefined0) - 1st hyper-fixed point of admissible

S[&sigma;](1undefined0undefined0undefined0) - 1st (&upsih;2)-order fixed point of admissible

S[&sigma;](1undefined0) - 1st (&upsih;&omega;)-order fixed point of admissible

S[&sigma;](1undefined0| 0) - 1st (&upsih;&upsih;)-order fixed point of admissible

S[&sigma;](1undefined0| 0| 0) - 1st (&upsih;&upsih; &upsih; )-order-fixed point of admissible

S[&sigma;&#39;1] - 1st inaccessible = П1-reflectingundefinedon П2-reflecting = 1st (&Delta;11-set of &upsih;)-order-fixed point of admissible, [KPi], [&Delta;12-CA+BI], &#123;zoo 2.3&#125; collapse&#123;zoo 1.22&#125;

S[&sigma;](S[&sigma;&#39;1]+1) - 1st admissible after inaccessible

S[&sigma;&#39;1](1) - 2nd inaccessible

S[&sigma;&#39;1](1undefined0) - 1st fixed point of inaccessible

S[&sigma;&#39;1](1undefined1) - 2nd fixed point of inaccessible

S[&sigma;&#39;1](1undefined0) - 1st 2-inaccessible = 1st (&Delta;11-set of &upsih;)-order-fixed point of inaccessible

S[&sigma;&#39;1](1undefined1) - 2nd 2-inaccessible

S[&sigma;&#39;1](1undefined1undefined0) - 1st fixed point of 2-inaccessible

S[&sigma;&#39;1](2undefined0) - 1st 3-inaccessible

S[&sigma;&#39;1](1undefined0undefined0) - 1st (1st fixed point of &alpha;)-inaccessible

S[&sigma;&#39;1](1undefined0undefined0) - 1st hyper-inaccessible, [KPh], &#123;zoo 2.4&#125;

S[&sigma;&#39;1](1undefined0undefined1) - 2nd hyper-inaccessible

S[&sigma;&#39;1](1undefined1undefined0) - 1st 2-hyper-inaccessible

S[&sigma;&#39;1](2undefined0undefined0) - 1st hyper2-inaccessible

S[&sigma;&#39;1](1undefined0undefined0undefined0) - 1st (&upsih;2)-order-inaccessible

S[&sigma;&#39;1](1undefined0) - 1st (&upsih;&omega;)-order-inaccessible

S[&sigma;&#39;1](1undefined0| 0) - 1st (&upsih;&upsih;)-order-inaccessible

S[&sigma;&#39;1](1undefined0| 0| 0) - 1st (&upsih;&upsih; &upsih; )-order-inaccessible

S[&sigma;&#39;2] - 1st Mahlo = П2-reflectingundefinedon П2-reflecting = 1st (&Delta;11-set of &upsih;)-order-inaccessible, [KPM], &#123;zoo 2.5&#125; collapse&#123;zoo 1.23&#125;

S[&sigma;](S[&sigma;&#39;2]+1) - 1st admissible after 1st Mahlo

S[&sigma;&#39;1](S[&sigma;&#39;2]+1) - 1st inaccessible after 1st Mahlo

S[&sigma;&#39;2](1) - 2nd Mahlo

S[&sigma;&#39;2](1undefined1) - 1st fixed point of Mahlo

S[&sigma;&#39;2](1undefined0) - 1st inaccessible limit of Mahlo

S[&sigma;&#39;2](1undefined1) - 2nd inaccessible limit of Mahlo

S[&sigma;&#39;2](1undefined1undefined0) - 1st fixed point of inaccessible limit of Mahlo

S[&sigma;&#39;2](2undefined0) - 2nd 2-inaccessible limit of Mahlo

S[&sigma;&#39;2](1undefined0undefined0) - 1st (1st fixed point of &alpha;)-inaccessible limit of Mahlo

S[&sigma;&#39;2](1undefined0undefined0) - 1st hyper-inaccessible limit of Mahlo

S[&sigma;&#39;2](1undefined0) - 1st Mahlo limit of Mahlo

S[&sigma;&#39;2](1undefined1) - 2nd Mahlo limit of Mahlo

S[&sigma;&#39;2](1undefined1undefined0) - 1st fixed point of Mahlo limit of Mahlo

S[&sigma;&#39;2](1undefined1undefined0) - 1st inaccessible limit of Mahlo limit of Mahlo

S[&sigma;&#39;2](1undefined1undefined0undefined0) - 1st hyper-inaccessible limit of Mahlo limit of Mahlo

S[&sigma;&#39;2](2undefined0) - 1st Mahlo limit of Mahlo limit of Mahlo = 1st Mahlo 2-limit of Mahlo

S[&sigma;&#39;2](1undefined0undefined0) - 1st Mahlo hyper-limit of Mahlo

S[&sigma;&#39;2](1undefined0) - 1st 2-Mahlo

S[&sigma;&#39;2](1undefined1) - 2nd 2-Mahlo

S[&sigma;&#39;2](1undefined1undefined0) - 1st fixed point of 2-Mahlo

S[&sigma;&#39;2](1undefined1undefined0) - 1st inaccessible limit of 2-Mahlo

S[&sigma;&#39;2](1undefined1undefined0) - 1st Mahlo limit of 2-Mahlo

S[&sigma;&#39;2](1undefined1undefined0) - 1st 2-Mahlo limit of 2-Mahlo

S[&sigma;&#39;2](2undefined0) - 1st 3-Mahlo

S[&sigma;&#39;2](2undefined1undefined0) - 1st Mahlo limit of 3-Mahlo

S[&sigma;&#39;2](2undefined1undefined0) - 1st 2-Mahlo limit of 3-Mahlo

S[&sigma;&#39;2](2undefined1undefined0) - 1st 3-Mahlo limit of 3-Mahlo

S[&sigma;&#39;2](&beta;undefined1undefined0) - 1st &gamma;-Mahlo limit of &beta;-Mahlo

S[&sigma;&#39;2](1undefined0undefined0) - 1st (1st fixed point of &alpha;)-Mahlo

S[&sigma;&#39;2](1undefined0undefined0) - 1st hyper-Mahlo

S[&sigma;&#39;2](1undefined0undefined0undefined0) - 1st (&upsih;2)-order-Mahlo

S[&sigma;&#39;2](1undefined0) - 1st (&upsih;&omega;)-order-Mahlo

S[&sigma;&#39;2](1undefined0| 0) - 1st (&upsih;&upsih;)-order-Mahlo

S[&sigma;&#39;3] - 1st П3-reflecting = 1st (&Delta;11-set of &upsih;)-order-Mahlo, [KP+П3-ref], &#123;zoo 2.6&#125; collapse&#123;zoo 1.24&#125;

S[&sigma;&#39;3](1) - 2nd П3-reflecting

S[&sigma;&#39;3](1undefined0) - 1st fixed point of П3-reflecting

S[&sigma;&#39;3](1undefined0) - 1st inaccessible limit of П3-reflecting

S[&sigma;&#39;3](1undefined0) - 1st Mahlo limit of П3-reflecting

S[&sigma;&#39;3](1undefined0) - 1st &alpha;-Mahlo limit of П3-reflecting

S[&sigma;&#39;3](1undefined0) - 1st П3-reflecting limit of П3-reflecting

S[&sigma;&#39;3](1undefined0) - 1st Mahlo in which П3-reflecting are stationary = 1st П3-reflecting (&Delta;11-set of &upsih;)-limit of Mahlo

S[&sigma;&#39;3](1undefined1undefined0) - 1st П3-reflecting limit of Mahlo in which П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined0) - 1st Mahlo in which П3-reflecting are stationary limit of Mahlo in which П3-reflecting are stationary

S[&sigma;&#39;3](2undefined0) - 1st 2-Mahlo in which П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined0) - 1st П3-reflecting limit of 2-Mahlo in which П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined0) - 1st Mahlo in which П3-reflecting are stationary limit of 2-Mahlo in which П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined0) - 1st 2-Mahlo in which П3-reflecting are stationary limit of 2-Mahlo in which П3-reflecting are stationary

S[&sigma;&#39;3](1undefined0) - 1st П3-reflecting in which П3-reflecting are stationary = 1st (&Delta;11-set of &upsih;)-order-Mahlo in which П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined0) - 1st П3-reflecting limit of П3-reflecting in which П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined0) - 1st Mahlo in which П3-reflecting are stationary limit of П3-reflecting in which П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined0) - 1st &alpha;-Mahlo in which П3-reflecting are stationary limit of П3-reflecting in which П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined0) - 1st П3-reflecting in which П3-reflecting are stationary limit of П3-reflecting in which П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined0) - 1st Mahlo in which (П3-reflecting in which П3-reflecting are stationary) are stationary

S[&sigma;&#39;3](1undefined1undefined1undefined0) - 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[&sigma;&#39;3](1undefined2undefined0) - 1st 2-Mahlo in which (П3-reflecting in which П3-reflecting are stationary) are stationary

S[&sigma;&#39;3](1undefined2undefined1undefined0) - 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[&sigma;&#39;3](2undefined) - 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[&sigma;&#39;3](2undefined1undefined0) - 1st П3-reflecting limit of П3-reflecting in which П3-reflecting are 2-stationary

S[&sigma;&#39;3](2undefined1undefined0) - 1st Mahlo in which П3-reflecting are stationary limit of П3-reflecting in which П3-reflecting are 2-stationary

S[&sigma;&#39;3](2undefined1undefined0) - 1st П3-reflecting in which П3-reflecting are stationary limit of П3-reflecting in which П3-reflecting are 2-stationary

S[&sigma;&#39;3](2undefined1undefined0) - 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[&sigma;&#39;3](3undefined) - 1st П3-reflecting in which П3-reflecting are 3-stationary

S[&sigma;&#39;3](1undefined0undefined0) - 1st П3-reflecting in which П3-reflecting are hyper-stationary

S[&sigma;&#39;3](1undefined) - 1st П3-reflecting onto П3-reflecting = 1st 2-П3-reflecting = 1st П3-reflecting in which П3-reflecting are (&Delta;11-set of &upsih;)-order-stationary

S[&sigma;&#39;3](1undefined1) - 2nd 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st fixed point of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st inaccessible limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st Mahlo limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st &alpha;-Mahlo limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st П3-reflecting limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st Mahlo in which П3-reflecting are stationary limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st &alpha;-Mahlo in which П3-reflecting are stationary limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st П3-reflecting in which П3-reflecting are &alpha;-stationary limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined11/3&#123;&#123;1&#125;&#125;0) - 1st 2-П3-reflecting limit of 2-П3-reflecting

S[&sigma;&#39;3](1undefined1undefined0) - 1st Mahlo in which 2-П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined1undefined0) - 1st Mahlo in which 2-П3-reflecting are stationary limit of Mahlo in which 2-П3-reflecting are stationary

S[&sigma;&#39;3](1undefined2undefined0) - 1st 2-Mahlo in which 2-П3-reflecting are stationary

S[&sigma;&#39;3](1undefined2undefined1undefined0) - 1st 2-Mahlo in which 2-П3-reflecting are stationary limit of 2-Mahlo in which 2-П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined) - 1st П3-reflecting in which 2-П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined1undefined0) - 1st П3-reflecting in which 2-П3-reflecting are stationary limit of П3-reflecting in which 2-П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined0) - 1st 2-П3-reflecting in which 2-П3-reflecting are stationary

S[&sigma;&#39;3](1undefined1undefined1undefined&#125;&#123;&#123;1&#125;&#125;| 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[&sigma;&#39;3](1undefined1undefined1undefined0) - 1st П3-reflecting in which (1st 2-П3-reflecting in which 2-П3-reflecting are stationary) are stationary

S[&sigma;&#39;3](1undefined1undefined1undefined1undefined&#125;&#123;&#123;1&#125;&#125;| 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[&sigma;&#39;3](1undefined1undefined1undefined1undefined0) - 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[&sigma;&#39;3](1undefined1undefined1undefined1undefined1undefined&#125;&#123;&#123;1&#125;&#125;| 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[&sigma;&#39;3](1undefined2undefined0) - 1st 2-П3-reflecting in which 2-П3-reflecting are 2-stationary

S[&sigma;&#39;3](1undefined2undefined1undefined&#125;&#123;&#123;1&#125;&#125;| 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[&sigma;&#39;3](2undefined0) - 1st 3-П3-reflecting

S[&sigma;&#39;3](2undefined1undefined0) - 1st П3-reflecting in which 3-П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined0) - 1st 2-П3-reflecting in which 3-П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined1undefined&#125;&#123;&#123;2&#125;&#125;| 0) - 1st П3-reflecting in which 3-П3-reflecting are stationary limit of 3-П3-reflecting in which 3-П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined0) - 1st 3-П3-reflecting in which 3-П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined1undefined0) - 1st П3-reflecting in which 3-П3-reflecting are stationary limit of 3-П3-reflecting in which 3-П3-reflecting are stationary

S[&sigma;&#39;3](2undefined1undefined1undefined&#125;&#123;&#123;2&#125;&#125;| 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[&sigma;&#39;3](2undefined1undefined1undefined&#125;&#123;&#123;2&#125;&#125;| 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[&sigma;&#39;3](2undefined1undefined1undefined1undefined0) - 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[&sigma;&#39;3](2undefined1undefined1undefined1undefined1undefined&#125;&#123;1&#123;2&#125;&#125;&#123;&#123;2&#125;&#125;| 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[&sigma;&#39;3](1undefined0undefined0) - 1st (1st fixed point of &alpha;)-П3-reflecting

S[&sigma;&#39;3](1undefined0undefined0) - 1st hyper-П3-reflecting

S[&sigma;&#39;3](1undefined0undefined0undefined0) - 1st (&upsih;2)-order-П3-reflecting

S[&sigma;&#39;3](1undefined0) - 1st (&upsih;&omega;)-order--П3-reflecting

S[&sigma;&#39;3](1undefined0| 0) - 1st (&upsih;&upsih;)-order-П3-reflecting

S[&sigma;&#39;3](1undefined10| 0| 0) - 1st (&upsih;&upsih; &upsih; )-order-П3-reflecting

S[&sigma;&#39;4] - 1st П4-reflecting = 1st (&Delta;11-set of &upsih;)-order-П3-reflecting, [KP+П4-ref]

S[&sigma;&#39;4](1undefined0) - 1st fixed point of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st inaccessible limit of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st Mahlo limit of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st &alpha;-Mahlo limit of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st П3-reflecting limit of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st П3-reflecting in which П3-reflecting are &alpha;-stationary limit of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st &alpha;-П3-reflecting limit of П4-reflecting

S[&sigma;&#39;4](1undefined&#125;&#123;&#123;&alpha;&#125;&#125;| 0) - 1st &gamma;-П3-reflecting in which &alpha;-П3-reflecting are &beta;-stationary limit of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st П4-reflecting limit of П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st Mahlo in which П4-reflecting are stationary

S[&sigma;&#39;4](1undefined1undefined0) - 1st Mahlo in which П4-reflecting are stationary limit of Mahlo in which П4-reflecting are stationary

S[&sigma;&#39;4](1undefined0) - 1st &alpha;-П3-reflecting in which П4-reflecting are stationary

S[&sigma;&#39;4](1undefined0) - 1st П4-reflecting in which П4-reflecting are stationary

S[&sigma;&#39;4](1undefined0) - 1st П3-reflecting that is П3-reflecting onto П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st П3-reflecting that is П3-reflecting onto П4-reflecting limit of П3-reflecting that is П3-reflecting onto П4-reflecting

S[&sigma;&#39;4](2undefined0) - 1st П3-reflecting that is 2-П3-reflecting onto П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st П4-reflecting that is П3-reflecting onto П4-reflecting

S[&sigma;&#39;4](2undefined0) - 1st П4-reflecting that is 2-П3-reflecting onto П4-reflecting

S[&sigma;&#39;4](1undefined0) - 1st П4-reflecting that is П4-reflecting onto П4-reflecting = 1st 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st П4-reflecting limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st Mahlo in witch П4-reflecting are stationary limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st П3-reflecting in witch П4-reflecting are &alpha;-stationary limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st П4-reflecting in which П4-reflecting are &alpha;-stationary limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st &alpha;-П3-reflecting that is П3-reflecting onto П4-reflecting limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined&#125;&#123;&#123;&alpha;&#125;&#125;/4| 0) - 1st &gamma;-П3-reflecting in which &alpha;-П3-reflecting are &beta;-stationary that is П3-reflecting onto П4-reflecting limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st П4-reflecting that is &alpha;-П3-reflecting onto П4-reflecting limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined&#125;&#123;&#123;&alpha;&#125;&#125;| 0) - 1st &gamma;-П3-reflecting in which П4-reflecting are &beta;-stationary that is &alpha;-П3-reflecting onto П4-reflecting limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st 2-П4-reflecting limit of 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st П4-reflecting in which 2-П4-reflecting are stationary

S[&sigma;&#39;4](1undefined1undefined0) - 1st &alpha;-П3-reflecting that is П3-reflecting onto П4-reflecting in which 2-П4-reflecting are stationary

S[&sigma;&#39;4](1undefined1undefined0) - 1st П4-reflecting that is &alpha;-П3-reflecting onto П4-reflecting in which 2-П4-reflecting are stationary

S[&sigma;&#39;4](1undefined1undefined0) - 1st 2-П4-reflecting in which 2-П4-reflecting are stationary

S[&sigma;&#39;4](1undefined1undefined0) - 1st П4-reflecting that is П3-reflecting onto 2-П4-reflecting

S[&sigma;&#39;4](1undefined2undefined0) - 1st П4-reflecting that is 2-П3-reflecting onto 2-П4-reflecting

S[&sigma;&#39;4](1undefined1undefined0) - 1st 2-П4-reflecting that is П3-reflecting onto 2-П4-reflecting

S[&sigma;&#39;5] - 1st П5-reflecting = 1st (&Delta;11-set of &upsih;)-order-П4-reflecting, [KP+П5-ref]

S[&sigma;&#39;5](1undefined0) - 1st fixed point of П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st inaccessible limit of П4-reflecting

S[&sigma;&#39;5](1undefined0) - 1st Mahlo limit of П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П3-reflecting limit of П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П4-reflecting limit of П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П4-reflecting limit of П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st Mahlo in which П5-reflecting are stationary

S[&sigma;&#39;5](1undefined0) - 1st П3-reflecting in which П5-reflecting are stationary

S[&sigma;&#39;5](1undefined0) - 1st П4-reflecting in which П5-reflecting are stationary

S[&sigma;&#39;5](1undefined0) - 1st П5-reflecting in which П5-reflecting are stationary

S[&sigma;&#39;5](1undefined0) - 1st П3-reflecting that is П3-reflecting onto П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П4-reflecting that is П3-reflecting onto П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П5-reflecting that is П3-reflecting onto П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П4-reflecting that is П4-reflecting onto П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П5-reflecting that is П4-reflecting onto П5-reflecting

S[&sigma;&#39;5](1undefined0) - 1st П5-reflecting that is П5-reflecting onto П5-reflecting = 2-П5-reflecting

S[&sigma;&#39;n] - 1st Пn-reflecting, [KP+Пn-ref]

S[&sigma;&#39;&omega;] = S[&sigma;+1] - (+1)-stable; L&sigma;≺1L&sigma;+1, [KP+П&omega;-ref], &#123;zoo 2.7&#125; collapse&#123;zoo 1.25&#125;

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st fixed point of (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st inaccessible limit of (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st Mahlo limit of (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st П3-reflecting limit of (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st (+1)-stable limit of (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st Mahlo in which (+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st П3-reflecting in which (+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st П4-reflecting in which (+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st (+1)-stable in which (+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st П3-reflecting that is П3-reflecting onto (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st П4-reflecting that is П3-reflecting onto (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st П5-reflecting that is П3-reflecting onto (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st (+1)-stable that is П3-reflecting onto (+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 2-(+1)-stable

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable in which (+1)-stable are &alpha;-stationary limit of 2-(+1)-stable

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П3-reflecting onto (+1)-stable limit of 2-(+1)-stable

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П4-reflecting onto (+1)-stable limit of 2-(+1)-stable

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П4-reflecting onto (+1)-stable limit of 2-(+1)-stable

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П3-reflecting onto (+1)-stable in which 2-(+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П4-reflecting onto (+1)-stable in which 2-(+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П5-reflecting onto (+1)-stable in which 2-(+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is (+1)-stable onto (+1)-stable in which 2-(+1)-stable are stationary

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П4-reflecting onto (+1)-stable that is П3-reflecting onto 2-П4-reflecting

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П5-reflecting onto (+1)-stable that is П3-reflecting onto 2-П4-reflecting

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is &alpha;-П6-reflecting onto (+1)-stable that is П3-reflecting onto 2-П4-reflecting

S[&sigma;&#39;&omega;](1undefined1undefined0) = S[&sigma;+1](1undefined1undefined0) - 1st (+1)-stable that is (+1)-stable onto (+1)-stable that is П3-reflecting onto 2-П4-reflecting

S[&sigma;&#39;&omega;](1undefined0undefined0) = S[&sigma;+1](1undefined0undefined0) - hyper-(+1)-stable

S[&sigma;&#39;&omega;](1undefined0undefined0undefined0) = S[&sigma;+1](1undefined0undefined0undefined0) - (&upsih;2)-order-(+1)-stable

S[&sigma;&#39;&omega;](1undefined0) = S[&sigma;+1](1undefined0) - 1st (&upsih;&omega;)-order-(+1)-stable

S[&sigma;&#39;&omega;](1undefined0| 0) = S[&sigma;+1](1undefined0| 0) - 1st (&upsih;&upsih;)-order-(+1)-stable

S[&sigma;&#39;&omega;+1] = S[&sigma;+1&#39;1] - (+1)-П1-reflecting (L&sigma;+1 ⊧ &phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;+1 ⊧ &phi;); &phi; is П1-formula); 1st (&Delta;11-set of &upsih;)-order-(+1)-stable

S[&sigma;&#39;&omega;+1](1undefined0) = S[&sigma;+1&#39;1](1undefined0) - 1st (+1)-П1-reflecting limit of (+1)-П1-reflecting

S[&sigma;&#39;&omega;+1](1undefined0) = S[&sigma;+1&#39;1](1undefined0) - 1st (+1)-П1-reflecting in which (+1)-П1-reflecting are stationary

S[&sigma;&#39;&omega;+1](1undefined0) = S[&sigma;+1&#39;1](1undefined0) - 1st (+1)-П1-reflecting that is П3-reflecting onto (+1)-П1-reflecting

S[&sigma;&#39;&omega;+1](1undefined0) = S[&sigma;+1&#39;1](1undefined0) - 1st (+1)-П1-reflecting that is (+1)-stable onto (+1)-П1-reflecting

S[&sigma;&#39;&omega;+1](1undefined0) = S[&sigma;+1&#39;1](1undefined0) - 2-(+1)-П1-reflecting

S[&sigma;&#39;&omega;+2] = S[&sigma;+1&#39;2] - (+1)-П2-reflecting (L&sigma;+1 ⊧ &phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;+1 ⊧ &phi;); &phi; is П2-formula)

S[&sigma;&#39;&omega;+n] = S[&sigma;+1&#39;n] - (+1)-Пn-reflecting (L&sigma;+1 ⊧ &phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;+1 ⊧ &phi;); &phi; is Пn-formula)

S[&sigma;&#39;&omega;&times;2] = S[&sigma;+2] - (+2)-stable; L&sigma;≺1L&sigma;+2

S[&sigma;&#39;&omega;&times;2+1] = S[&sigma;+2&#39;1] - (+2)-П1-reflecting (L&sigma;+2 ⊧ &phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;+2 ⊧ &phi;); &phi; is П1-formula)

S[&sigma;&#39;&omega;&times;2+2] = S[&sigma;+2&#39;2] - (+2)-П2-reflecting (L&sigma;+2 ⊧ &phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;+2 ⊧ &phi;); &phi; is П2-formula)

S[&sigma;&#39;&omega;&times;2+n] = S[&sigma;+2&#39;n] - (+2)-Пn-reflecting (L&sigma;+2 ⊧ &phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;+2 ⊧ &phi;); &phi; is Пn-formula)

S[&sigma;&#39;&omega;&times;3] = S[&sigma;+3] - (+3)-stable; L&sigma;≺1L&sigma;+3

S[&sigma;&#39;&omega;2] = S[&sigma;+&omega;] = S[&sigma;+S[&sigma;]] - (+&omega;)-stable; L&sigma;≺1L&sigma;+&omega;

S[&sigma;&#39;&omega;3] = S[&sigma;+&omega;&times;2] - (+&omega;&times;2)-stable; L&sigma;≺1L&sigma;+&omega;&times;2

S[&sigma;&#39;&omega;&omega;] = S[&sigma;+&omega;2] - (+&omega;2)-stable; L&sigma;≺1L&sigma;+&omega;2

S[&sigma;&#39;&omega;&omega; &omega; ] = S[&sigma;+&omega;&omega;] - (+&omega;&omega;)-stable; L&sigma;≺1L&sigma;+&omega;&omega;

S[&sigma;&#39;&epsilon;0] = S[&sigma;+&epsilon;0] - (+&epsilon;0)-stable; L&sigma;≺1L&sigma;+&epsilon; 0

S[&sigma;&#39;S[&sigma;](1)] = S[&sigma;+S[&sigma;](1)] - (+&omega;1CK)-stable; L&sigma;≺1L&omega; 1CK

S[&sigma;&#39;S[&sigma;&#39;&omega;]] = S[&sigma;+S[&sigma;+1]] - (+(+1)-stable)-stable; L&sigma;≺1L&sigma;+L &beta;≺L&beta;+1

S[&sigma;+&alpha;] - (+&alpha;)-stable; L&sigma;≺1L&sigma;+&alpha;

S[&sigma;&times;2] = S[&sigma;+&sigma;] - &sigma;=(+&sigma;)-stable; L&sigma;≺1L&sigma;+&sigma; [KPi+&forall;n&exist;&sigma;&ge;n(L&sigma;≺1L&sigma;+n)], collapse&#123;zoo 1.26&#125;

S[&sigma;&times;2+1] - &sigma;=(+&sigma;+1)-stable

S[&sigma;&times;2+&alpha;] - &sigma;=(+&sigma;+&alpha;)-stable

S[&sigma;&times;3] - &sigma;=(+&sigma;&times;2)-stable

S[&sigma;&times;&omega;] - &sigma;=(+&sigma;&times;&omega;)-stable

S[&sigma;&times;&alpha;] - &sigma;=(+&sigma;&times;&alpha;)-stable

S[&sigma;2] - &sigma;=(+&sigma;&times;&sigma;)-stable

S[&sigma;&omega;] - &sigma;=(+&sigma;&omega;)-stable

S[&sigma;&alpha;] - &sigma;=(+&sigma;&alpha;)-stable

S[&sigma;&sigma;] - &sigma;=(+&sigma;&sigma;)-stable

S[&epsilon;&sigma;+1] - &sigma;=(+&epsilon;&sigma;+1)-stable

S[S[&sigma;2](&sigma;+1)] - &sigma;=(&Delta;11-set of &sigma;)-stable; (&sigma;+)-stable; (next admissible)-stable [KP+П11-ref], &#123;zoo 2.8&#125;

S[S[&sigma;2](&sigma;+1)](1undefined](&sigma;+1)| ) - 1st (&sigma;+)-stable limit of (&sigma;+)-stable

S[S[&sigma;2](&sigma;+1)](1undefined](&sigma;+1)| ) - 1st (&sigma;+)-stable in which (&sigma;+)-stable are stationary

S[S[&sigma;2](&sigma;+1)](1undefined](&sigma;+1)| ) - 1st (&sigma;+)-stable that is П3-reflecting onto (&sigma;+)-stable

S[S[&sigma;2](&sigma;+1)](1undefined](&sigma;+1)| ) - 1st (&sigma;+)-stable that is (+1)-stable onto (&sigma;+)-stable

S[S[&sigma;2](&sigma;+1)](1undefined](&sigma;+1)| ) - 1st (&sigma;+)-stable that is &sigma;=(+&sigma;)-stable onto (&sigma;+)-stable

S[S[&sigma;2](&sigma;+1)](1undefined](&sigma;+1)| ) - 2-(&sigma;+)-stable

S[S[&sigma;2](&sigma;+1)&#39;1] - (&sigma;+)-П1-reflecting

S[S[&sigma;2](&sigma;+1)+1] - (&sigma;++1)-stable

S[S[&sigma;2](&sigma;+1)+&alpha;] - (&sigma;++&alpha;)-stable

S[S[&sigma;2](&sigma;+1)+&sigma;] - (&sigma;++&sigma;)-stable

S[S[&sigma;2](&sigma;+1)+S[&sigma;2](&sigma;+1)] - (&sigma;+&times;2)-stable

S[S[&sigma;2](&sigma;+1)S[&sigma;2](&sigma;+1)] - (&sigma;+&sigma; + )-stable

S[&epsilon;S[&sigma; 2](&sigma;+1) ] - (&epsilon;&sigma;++1)-stable

S[S[&sigma;2](&sigma;+2)] - &sigma;=(&Delta;11-set of &sigma;+)-stable; (&sigma;++)-stable; (next 2nd admissible)-stable &#123;zoo 2.10&#125;

S[S[&sigma;2](&sigma;+2)](1undefined](&sigma;+2)| ) - 1st (&sigma;++)-stable limit of (&sigma;++)-stable

S[S[&sigma;2](&sigma;+2)](1undefined](&sigma;+2)| ) - 1st (&sigma;++)-stable in which (&sigma;++)-stable are stationary

S[S[&sigma;2](&sigma;+2)](1undefined](&sigma;+2)| ) - 1st (&sigma;++)-stable that is П3-reflecting onto (&sigma;++)-stable

S[S[&sigma;2](&sigma;+2)](1undefined](&sigma;+2)| ) - 1st (&sigma;++)-stable that is (+1)-stable onto (&sigma;++)-stable

S[S[&sigma;2](&sigma;+2)](1undefined](&sigma;+2)| ) - 1st (&sigma;++)-stable that is &sigma;=(+&sigma;)-stable onto (&sigma;++)-stable

S[S[&sigma;2](&sigma;+2)](1undefined](&sigma;+1)/S[&sigma;2](&sigma;+2)| ) - 1st (&sigma;++)-stable that is (&sigma;+)-stable onto (&sigma;++)-stable

S[S[&sigma;2](&sigma;+2)](1undefined](&sigma;+2)| ) - 2-(&sigma;++)-stable

S[S[&sigma;2](&sigma;+3)] - &sigma;=(&Delta;11-set of &sigma;++)-stable; (&sigma;+++)-stable; (next 3d admissible)-stable

S[S[&sigma;2](&sigma;+&omega;)] - (next &omega;-th admissible)-stable [limit of DAN]

S[S[&sigma;2](&sigma;+&alpha;)] - (next &alpha;-th admissible)-stable

S[S[&sigma;2](&sigma;+&sigma;)] - (next &sigma;=(&sigma;-th admissible))-stable

S[S[&sigma;2](&sigma;&sigma;)] - (next &sigma;=(&sigma;&sigma;-th admissible))-stable

S[S[&sigma;2](S[&sigma;2](&sigma;+1))] - (next &sigma;=(&sigma;+-th admissible))-stable; (next next admissible)-stable

S[S[&sigma;2](S[&sigma;2](&sigma;+2))] - (next &sigma;=(&sigma;++-th admissible))-stable; (next next 2nd admissible)-stable

S[S[&sigma;2](S[&sigma;2](S[&sigma;2](&sigma;+1)))] - (next next next admissible)-stable

S[S[&sigma;2](1undefined| &sigma;+1)]- (limit of next admissible)-stable

S[S[&sigma;2]([S[&sigma;2](1undefined| &sigma;+1)+1)]- (next after limit of next admissible)-stable

S[S[&sigma;2]([S[&sigma;2](1undefined| &sigma;+1)+2)]- (next 2nd after limit of next admissible)-stable

S[S[&sigma;2](S[&sigma;2]([S[&sigma;2](1undefined| &sigma;+1)+1))]- (next next after limit of next admissible)-stable

S[S[&sigma;2](1undefined| &sigma;+2)] - (2nd limit of next admissible)-stable

S[S[&sigma;2](2undefined| &sigma;+1)] - (2-limit of next admissible)-stable

S[S[&sigma;2](1undefined| 0undefined| &sigma;+1)] - (hyper-limit of next admissible)-stable

S[S[&sigma;2](1undefined| 0undefined| 0undefined| &sigma;+1)] - ((&upsih;&omega;)-order-limit of next admissible)-stable

S[S[&sigma;2&#39;1](&sigma;+1)] - (next inaccessible)-stable &#123;zoo 2.11&#125;

S[S[&sigma;2&#39;1](&sigma;+1)+1] - (next inaccessible+1)-stable

S[S[&sigma;2](S[&sigma;2&#39;1](&sigma;+1)+1)] - (next admissible after next inaccessible+1)-stable

S[S[&sigma;2&#39;1](&sigma;+2)] - (2nd next inaccessible)-stable

S[S[&sigma;2&#39;1](S[&sigma;2&#39;1](&sigma;+1))] - (next next inaccessible)-stable

S[S[&sigma;2&#39;1](1undefined| &sigma;+1)] - (1st limit of next inaccessible)-stable

S[S[&sigma;2&#39;1](1undefined&#39;1| &sigma;+1)] - (1st next 2-inaccessible)-stable

S[S[&sigma;2&#39;1](1undefined&#39;1| 0undefined&#39;1| &sigma;+1)] - (1st next hyper-inaccessible)-stable

S[S[&sigma;2&#39;1](1undefined&#39;1| 0undefined&#39;1| 0undefined&#39;1| &sigma;+1)] - ((&upsih;&omega;)-order-inaccessible)-stable

S[S[&sigma;2&#39;2](&sigma;+1)] - (next Mahlo)-stable &#123;zoo 2.12&#125;

S[S[&sigma;2&#39;2](&sigma;+1)] - (next П3-reflecting)-stable

S[S[&sigma;2&#39;n](&sigma;+1)] - (next Пn-reflecting)-stable

S[S[&sigma;2+1](&sigma;+1)] - (next (+1)-stable)-stable; doubly (+1)-stable; L&sigma;≺1L&beta;≺1L&beta;+1

S[S[&sigma;2+1](&sigma;+1)+1] - (next (+1)-stable+1)-stable

S[S[&sigma;2+1](&sigma;+2)] - (next 2nd (+1)-stable)-stable

S[S[&sigma;2+1](S[&sigma;2+1](&sigma;+1))] - (next next (+1)-stable)-stable

S[S[&sigma;2+1](1undefined| &sigma;+1)] - (1st limit of next (+1)-stable)-stable

S[S[&sigma;2+1](1undefined&#39;1| &sigma;+1)] - (1st next inaccessible limit of next (+1)-stable)-stable)

S[S[&sigma;2+1](1undefined&#39;1/&sigma;2&#39;2| &sigma;+1)] - (1st next Mahlo limit of next (+1)-stable)-stable)

S[S[&sigma;2+1](1undefined&#39;1/&sigma;2+1| &sigma;+1)] - (1st next (+1)-stable limit of next (+1)-stable)-stable)

S[S[&sigma;2+1](1undefined&#39;1| &sigma;+1)] - (1st next Mahlo that is next (+1)-stable are stationary)-stable)

S[S[&sigma;2+1](1undefined+1| &sigma;+1)] - (1st next 2-(+1)-stable are stationary)-stable)

S[S[&sigma;2+1&#39;1](1undefined+1| &sigma;+1)] - (1st next (+1)-П1-reflecting)-stable)

S[S[&sigma;2+2](&sigma;+1)] - (next (+2)-stable)-stable; doubly (+2)-stable; L&sigma;≺1L&beta;≺1L&beta;+2 &#123;zoo 2.13&#125;

S[S[&sigma;2+&alpha;](&sigma;+1)] - (next (+&alpha;)-stable)-stable; doubly (+&alpha;)-stable

S[S[&sigma;2+&sigma;2](&sigma;+1)] - (next &sigma;=(+&sigma;)-stable)-stable; doubly (+&sigma;2)-stable

S[S[&epsilon;&sigma; 2+1 ](&sigma;+1)] - (next &sigma;=(+&epsilon;&sigma;+1)-stable)-stable; doubly (+&epsilon;&sigma;+1)-stable

S[S[S[&sigma;3](&sigma;2+1)](&sigma;+1)] - (next (next admissible)-stable)-stable

S[S[S[&sigma;3](&sigma;2+1)](&sigma;+1)+1] - (next (next admissible)-stable+1)-stable

S[S[S[&sigma;3](&sigma;2+1)](&sigma;+1)+1] - (next (next admissible)-stable+1)-stable

S[S[S[&sigma;3](&sigma;2+1)](&sigma;+2)] - (next 2nd (next admissible)-stable)-stable

S[S[S[&sigma;3](&sigma;2+1)+1](&sigma;+1)] - (next ((next admissible)+1)-stable)-stable

S[S[S[&sigma;3](&sigma;2+2)](&sigma;+1)] - (next (next 2nd admissible)-stable)-stable

S[S[S[&sigma;3](&sigma;2+1)](&sigma;+1)] - (next (next admissible)-stable)-stable

S[S[S[&sigma;3](S[&sigma;3](&sigma;2+1))](&sigma;+1)] - (next (next next admissible)-stable)-stable

S[S[S[&sigma;3](1undefined| &sigma;2+1)](&sigma;+1)] - (next (limit of next admissible)-stable)-stable

S[S[S[&sigma;3&#39;1](&sigma;2+1)](&sigma;+1)] - (next (next inaccessible)-stable)-stable

S[S[S[&sigma;3&#39;2](&sigma;2+1)](&sigma;+1)] - (next (next Mahlo)-stable)-stable

S[S[S[&sigma;3&#39;2](&sigma;2+1)](&sigma;+1)] - (next (next П3-reflecting)-stable)-stable

S[S[S[&sigma;3&#39;n](&sigma;2+1)](&sigma;+1)] - (next (next Пn-reflecting)-stable)-stable

S[S[S[&sigma;3+1](&sigma;2+1)](&sigma;+1)] - (next (next (+1)-stable)-stable)-stable; triply (+1)-stable; L&sigma;≺1L&beta;≺1L&gamma;≺1L&gamma;+1

S[S[S[S[&sigma;4+1](&sigma;3+1)](&sigma;2+1)](&sigma;+1)] - (next (next (next (+1)-stable)-stable)-stable)-stable; quadruply (+1)-stable; L&sigma;≺1L&beta;≺1L&gamma;≺1L&delta;≺1L&delta;+1

S[...S[&sigma;n+1]...(&sigma;+1)] - n-ply (+1)-stable;

S[S[&sigma;&omega;]] - &omega;-ply stable; nonprojectable; strongly &Sigma;1-admissible, [П12-CA0], [&Delta;13-CA0], &#123;zoo 2.15&#125;

S[S[&sigma;&omega;]](1) - 2nd &omega;-ply stable

S[S[&sigma;&omega;]](&omega;) - &omega;-th &omega;-ply stable

S[S[&sigma;&omega;]&#39;1] - (&omega;-ply stable)-П1-reflecting

S[S[&sigma;&omega;]+1] - (next (...(&omega;-ply stable)...)-stable+1)-stable

S[S[&sigma;&omega;]+1](1) - 2nd (next (...(&omega;-ply (+1)-stable)...)-stable+1)-stable

S[S[S[&sigma;&omega;](&sigma;+1)+1]] - (next (next (...(&omega;-ply (+1)-stable)...)-stable+1)-stable)-stable

S[S[S[&sigma;&omega;](&sigma;+2)]] - (next (2nd next (...(&omega;-ply (+1)-stable)...)-stable)-stable)-stable

S[S[S[S[&sigma;&omega;](&sigma;2+1)+1](&sigma;+1)]] - (next (next (next (...(&omega;-ply (+1)-stable)...)-stable+1)-stable)-stable)-stable

S[S[S[S[&sigma;&omega;](&sigma;2+2)](&sigma;+1)]] - (next (next (2nd next (...(&omega;-ply (+1)-stable)...)-stable)-stable)-stable)-stable

S[S[&sigma;&omega;](1)] - nonprojectable limit of nonprojectables

S[S[&sigma;&omega;](&omega;)] - &omega;-th nonprojectable limit of nonprojectables

S[S[&sigma;&omega;&#39;1]] - nonprojectable and admissible

S[S[&sigma;&omega;&#39;1](1)] - nonprojectable limit of (nonprojectable and admissible)

S[S[&sigma;&omega;&#39;1](2)] - 2nd nonprojectable limit of (nonprojectable and admissible)

S[S[&sigma;&omega;&#39;2]] - nonprojectable and Mahlo

S[S[&sigma;&omega;&#39;2](1)] - nonprojectable limit of (nonprojectable and Mahlo)

S[S[&sigma;&omega;&#39;2](2)] - 2nd nonprojectable limit of (nonprojectable and Mahlo)

S[S[&sigma;&omega;&#39;3]] - nonprojectable and П3-reflecting

S[S[&sigma;&omega;&#39;3](1)] - nonprojectable limit of (nonprojectable and П3-reflecting)

S[S[&sigma;&omega;&#39;3](2)] - 2nd nonprojectable limit of (nonprojectable and П3-reflecting)

S[S[&sigma;&omega;&#39;n]] - nonprojectable and Пn-reflecting

S[S[&sigma;&omega;&#39;n](1)] - nonprojectable limit of (nonprojectable and Пn-reflecting)

S[S[&sigma;&omega;+1]] = S[S[&sigma;&omega;+1]] - nonprojectable and (+1)-stable; (&omega;+1)-ply (+1)-stable [KP+&Sigma;1-sep], [П12-CA+BI]

S[S[&sigma;&omega;+1+1]+1] - (next (...((&omega;+1)-ply (+1)-stable)...)-stable+1)-stable

S[S[S[&sigma;&omega;+1+1](&sigma;+1)+1]] - (next (next (...((&omega;+1)-ply (+1)-stable)...)-stable+1)-stable)-stable

S[S[S[S[&sigma;&omega;+1+1](&sigma;2+1)+1](&sigma;+1)]] - (next (next (next (...((&omega;+1)-ply (+1)-stable)...)-stable+1)-stable)-stable)-stable

S[S[&sigma;&omega;](S[&sigma;&omega;+1+1]+1)] - nonprojectable limit of (&omega;+1)-ply (+1)-stable

S[S[&sigma;&omega;&#39;1](S[&sigma;&omega;+1+1]+1)] - nonprojectable limit of (nonprojectable limit of (&omega;+1)-ply (+1)-stable and admissible)

S[S[&sigma;&omega;&#39;2](S[&sigma;&omega;+1+1]+1)] - nonprojectable limit of (nonprojectable limit of (&omega;+1)-ply (+1)-stable and Mahlo)

S[S[&sigma;&omega;&#39;n](S[&sigma;&omega;+1+1]+1)] - nonprojectable limit of (nonprojectable limit of (&omega;+1)-ply (+1)-stable and Пn-reflecting)

S[S[&sigma;&omega;+1+2]] - nonprojectable and (+2)-stable; (&omega;+1)-ply (+2)-stable

S[S[&sigma;&omega;+1+&sigma;&omega;+1]] - nonprojectable and &sigma;=(+&sigma;)-stable; (&omega;+1)-ply &sigma;=(+&sigma;)-stable

S[S[&sigma;&omega;+2](&sigma;&omega;+1+1)] - nonprojectable and (next admissible)-stable; (&omega;+1)-ply (next admissible)-stable

S[S[&sigma;&omega;+2&#39;1](&sigma;&omega;+1+1)] - nonprojectable and (next inaccessible)-stable; (&omega;+1)-ply (next inaccessible)-stable

S[S[&sigma;&omega;+2&#39;2](&sigma;&omega;+1+1)] - nonprojectable and (next Mahlo)-stable; (&omega;+1)-ply (next Mahlo)-stable

S[S[&sigma;&omega;+2]] - nonprojectable and doubly (+1)-stable; (&omega;+2)-ply (+1)-stable

S[S[&sigma;&omega;+3]] - nonprojectable and triply (+1)-stable; (&omega;+3)-ply (+1)-stable

S[S[&sigma;&omega;&times;2]] - doubly nonprojectable; (&omega;&times;2)-ply stable

S[S[&sigma;&omega;&times;2+1] = S[S[&sigma;&omega;&times;2+1]] - doubly nonprojectable and (+1)-stable; (&omega;&times;2+1)-ply stable

S[S[&sigma;&epsilon; 0 ]] - &epsilon;0-ply stable [&Delta;13-CA]

S[S[&sigma;&beta;+1] = S[S[&sigma;&beta;+1]] - (&beta;+1)-ply stable, where n - limit ordinal

S[S[&sigma;&alpha;]] - &alpha;-ply stable

Up to Z2. Next comes a speculative guess.

S[S[&sigma;&sigma;]] - &sigma;=(&sigma;-ply stable)

S[S[&sigma;&sigma;+1]] - &sigma;=(&sigma;+1-ply stable)

S[S[&sigma;&epsilon; &sigma;+1 ]] - &sigma;=(&epsilon;&sigma;+1-ply stable)

S[S[&sigma;S[&sigma; 2](&sigma;+1) ]] - (next admissible)-ply stable

S[S[&sigma;S[&sigma; 2+1](&sigma;+1) ]] - ((+1)-stable)-ply stable

S[S[&sigma;S[&sigma; &omega;] ]] - (next nonprojectable)-ply stable

S[S[&sigma;&sigma; 2 ]] = S[S[&sigma;S[&sigma; &sigma;] ]] - &sigma;=((&sigma;-ply stable)-ply stable); doubly-ply stable

S[S[&sigma;S[&sigma; S[&sigma;2 ](&sigma;+1)] ]] - (((+1)-stable)-ply stable)-ply stable

S[S[&sigma;&sigma; 3 ]] = S[S[&sigma;S[&sigma; S[&sigma;&sigma; ]] ]] - &sigma;=(((&sigma;-ply stable)-ply stable)-ply stable); triply-ply stable

S[S[&sigma;&sigma; n ]] = n-ply-ply stable

S[S[&sigma;&sigma; &sigma; ]] = &sigma;=(&sigma;-ply-ply stable)

S[S[&sigma;&sigma; &sigma; 2 ]] = doubly-ply-ply stable

S[S[&sigma;&sigma; &sigma; n ]] = n-ply-ply-ply stable

Add a little inductivity to notation. Let:

S[&sigma;] = S[S2[&sigma;&sigma;]]

S[&sigma;2] = S[S2[&sigma;&sigma;](1)]

S[&sigma;&omega;] = S[S2[&sigma;&sigma;](&omega;)]

S[&sigma;&sigma;] = S[S2[&sigma;&sigma;](S2[&sigma;&sigma;])]

S[&sigma;&sigma; &sigma; ] = S[S2[&sigma;&sigma;](S2[&sigma;&sigma;](S2[&sigma;&sigma;])))]

e.t.c.

Then we get the following extension:

S[S[S2[&sigma;&sigma;](1undefined0)]] = S[S2[&sigma;&sigma;](S2[&sigma;&sigma;](S2[&sigma;&sigma;](...))))] - limit of ...-ply stable

S[S[S2[&sigma;&sigma;](1undefined1)]] - 2nd limit of ...-ply stable

S[S[S2[&sigma;&sigma;](2undefined0)]] - 2-limit of ...-ply stable

S[S[S2[&sigma;&sigma;](1undefined0undefined0)]] - hyper-limit of ...-ply stable

S[S[S2[&sigma;&sigma;&#39;1]]] - П2-(St)-reflecting, where St - set of stable below ; inaccessible limit of ...-ply stable, [KP+П1-сoll], [KP+&Delta;2-sep], [&Delta;13-CA+BI], &#123;zoo 2.16&#125;

S[S[S2[&sigma;&sigma;&#39;2]]] - П2-(St)-reflecting on П2-(St)-reflecting; Mahlo limit of ...-ply stable

S[S[S2[&sigma;&sigma;&#39;3]]] - П3-(St)-reflecting; П3-reflecting limit of ...-ply stable

S[S[S2[&sigma;&sigma;&#39;n]]] - Пn-(St)-reflecting; Пn-reflecting limit of ...-ply stable

S[S[S2[&sigma;&sigma;+1]]] - (+1)-2-stable; L&sigma;&sigma;≺2LSt+1

S[S[S2[&sigma;&sigma;+1]]](1) -2nd ((+1)-2-stable)-stable

S[S[S2[&sigma;&sigma;+1]]+1] - (next (...((+1)-2-stable)...)-stable+1)-stable

S[S[S[S2[&sigma;&sigma;+1]](&sigma;+1)+1]] - (next (next (...((+1)-2-stable)...)-stable+1)-stable)-stable

S[S[S[S[S2[&sigma;&sigma;+1]](&sigma;2+2)](&sigma;+1)+1]]] - (next (next (next (...((+1)-2-stable)...)-stable+1)-stable)-stable)-stable

S[S[S[S2[&sigma;&sigma;+1]](&sigma;&omega;)](1)] - (next nonprojectable limit of nonprojectables after (+1)-2-stable))-stable

S[S[S[S2[&sigma;&sigma;+1]](&sigma;&sigma;)](1)] - (next &sigma;=(&sigma;-ply stable) limit of nonprojectables after (+1)-2-stable))-stable

S[S[S[S2[&sigma;&sigma;+1]](S2[&sigma;&sigma;&#39;1])](1)] - (next П2-(St)-reflecting limit of nonprojectables after (+1)-2-stable))-stable

S[S[S2[&sigma;&sigma;+1]](1)] - (+1)-2-stable limit of nonprojectables

S[S[S2[&sigma;&sigma;+1]+1]] - ((+1)-2-stable+1)-stable

S[S[S[&sigma;](S2[&sigma;&sigma;+1]+1)]] - (next admissible after (+1)-2-stable)-stable

S[S[S[S[&sigma;2+1](&sigma;+1)](S2[&sigma;&sigma;+1]+1)]] - (next doubly (+1)-stable after (+1)-2-stable)-stable

S[S[S[&sigma;&omega;](S2[&sigma;&sigma;+1]+1)]] - (next nonprojectable after (+1)-2-stable)-stable

S[S[S2[&sigma;&sigma;+1](1)]] - (2nd (+1)-2-stable)-stable

S[S[S2[&sigma;&sigma;+1&#39;1]]] - (+1)-П1-(St)-reflecting = ((&Delta;11-set of &upsih;)-order-(+1)-2-stable)-stable

S[S[S2[&sigma;&sigma;+1&#39;2]]] - (+1)-П2-(St)-reflecting

S[S[S2[&sigma;&sigma;+2]]] - (+2)-2-stable; L&sigma;&sigma;≺2LSt+2

S[S[S2[&sigma;&sigma;+&sigma;&sigma;]]] - &sigma;=(+&sigma;)-2-stable; L&sigma;&sigma;≺2LSt+&sigma;&sigma;

S[S[S2[S[&sigma;](&sigma;&sigma;+1)]]] - (next admissible)-2-stable

S[S[S2[S[&sigma;+1](&sigma;&sigma;+1)]]] - (next (+1)-stable)-2-stable

S[S[S2[S[&sigma;2+1](S[&sigma;+1](&sigma;&sigma;+1))]]] - (next (next (+1)-stable)-stable)-2-stable

S[S[S2[S[S[&sigma;&omega;]](&sigma;&sigma;+1)]]] - (next nonprojectable)-2-stable

S[S[S2[S[S[&sigma;&sigma;]](&sigma;&sigma;+1)]]] - (next &sigma;=(&sigma;-ply stable))-2-stable

S[S[S2[S[S[S2[&sigma;&sigma;2&#39;1]]](&sigma;&sigma;+1)]]] - (next П2-(St)-reflecting)-2-stable

S[S[S2[S[S[S2[&sigma;&sigma;2+1]]](&sigma;&sigma;+1)]]] - (next 2-stable)-2-stable = doubly (+1)-2-stable; L&sigma;&sigma;≺2 L&beta;≺2LSt+1

S[S[S2[S[S[S2[S[S[S2[&sigma;&sigma;3&#39;1]]](&sigma;&sigma;2+1)]]](&sigma;&sigma;+1)]]] - (next (next 2-stable)-2-stable)-2-stable = triply (+1)-2-stable; L&sigma;&sigma;≺2 L&beta;≺ L&gamma;≺2LSt+1

S[S[S2[S[S[S2[&sigma;&sigma;&omega;]]]]]] - &omega;-ply 2-stable; 2-nonprojectable; strongly &Sigma;2-admissible [П13-CA0], [&Delta;14-CA0]

S[S[S2[S[S[S2[&sigma;&sigma;&omega;+1]]]]]] - (&omega;+1)-ply 2-stable [KP+&Sigma;2-sep], [П13-CA+BI]

S[S[S2[S[S[S2[&sigma;&sigma;&epsilon; 0 ]]]]]] - (&epsilon;0)-ply 2-stable [&Delta;14-CA]

S[S[S2[S[S[S2[&sigma;&sigma;&sigma;&sigma;]]]]]] - &sigma;=(&sigma;-ply 2-stable)

S[S[S2[S[S[S2[&sigma;&sigma;&sigma;&sigma; &sigma;&sigma; ]]]]]] = &sigma;=(&sigma;-ply-ply 2-stable)

S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;](1undefined0)]]]]]]] - limit of ...-ply 2-stable

S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;&#39;1]]]]]]] - П2-(St2)-reflecting, where St2 - set of 2-stable below ; inaccessible limit of ...-ply 2-stable, [KP+П2-сoll], [KP+&Delta;3-sep], [&Delta;14-CA+BI]

S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;+1]]]]]]] - (+1)-3-stable; L&sigma;&sigma;≺3LSt 2+1

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;&omega;]]]]]]]]]]]]] - &omega;-ply 3-stable; 3-nonprojectable; strongly &Sigma;3-admissible [П14-CA0], [&Delta;15-CA0]

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;&omega;+1]]]]]]]]]]]]] - (&omega;+1)-ply 3-stable [KP+&Sigma;3-sep], [П13-CA+BI]

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;&epsilon; 0 ]]]]]]]]]]]]] - (&epsilon;0)-ply 3-stable [&Delta;15-CA]

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[&sigma;&sigma;&sigma;&sigma;+1]]]]]]]]]]]]]] - П2-(St3)-reflecting, where St3 - set of 3-stable below ; inaccessible limit of ...-ply 3-stable, [KP+П3-сoll], [KP+&Delta;4-sep], [&Delta;15-CA+BI]

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[&sigma;&sigma;&sigma;&sigma;&omega;]]]]]]]]]]]]]]]]]]]]]]]]]]]] - &omega;-ply 4-stable; 4-nonprojectable; strongly &Sigma;4-admissible [П15-CA0], [&Delta;16-CA0]

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[&sigma;&sigma;&sigma;&sigma;&omega;+1]]]]]]]]]]]]]]]]]]]]]]]]]]]] - (&omega;+1)-ply 4-stable [KP+&Sigma;4-sep], [П14-CA+BI]

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[&sigma;&sigma;&sigma;&sigma;&epsilon; 0 ]]]]]]]]]]]]]]]]]]]]]]]]]]]] - (&epsilon;0)-ply 4-stable [&Delta;16-CA]

S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[S4[S5[&sigma;&sigma;&sigma;&sigma;&sigma;+1]]]]]]]]]]]]]]]]]]]]]]]]]]]]] - П2-(St4)-reflecting, where St4 - set of 4-stable below ; inaccessible limit of ...-ply 4-stable, [KP+П4-сoll], [KP+&Delta;5-sep], [&Delta;16-CA+BI]

Up to ZFC. Next comes a very speculative guess.

Let&#39;s add more inductivity to notation.

S[&sigma;] = SG[g][G[g]]

S[&sigma;2] = SG[g][G[g](1)]

S2[&sigma;&sigma;] = SG[g&#39;1][G[g&#39;1]]

S2[&sigma;&sigma;2] = SG[g&#39;1][G[g&#39;1](1)]

S3[&sigma;&sigma;&sigma;] = SG[g&#39;2][G[g&#39;2]]

S3[&sigma;&sigma;&sigma;2] = SG[g&#39;2][G[g&#39;2](1)]

e.t.c.

Then we get the following extension:

S[S&omega;[&sigma;(&omega;)]] = S[SG[g+1][G[g+1]]] = G[g+1] - start 1st 2nd-order gap length 1 and (+1)-stable; (L&beta;/L&beta;+1)&cap;&omega;1=&empty; [Z2], [ZFC-],  &#123;zoo 2.17&#125;

S[S&omega;[&sigma;(&omega;)]+1] = S[SG[g+1][G[g+1]]+1] - start 1st 2nd-order gap length 1 and (+2)-stable

S[S&omega;+1[&sigma;(&omega;+1)]] = S[SG[g+1&#39;1][G[g+1&#39;1]]] - start 1st 2nd-order gap length 1 and (+1)-2-stable

S[S&omega;+1[&sigma;(&omega;+1)]+1] = S[SG[g+1&#39;1][G[g+1&#39;1]]+1] - start 1st 2nd-order gap length 1 and (+2)-2-stable

S[S&omega;&times;2[&sigma;(&omega;&times;2)]] = S[SG[g+2][G[g+2]]] - start 1st 2nd-order gap length 2 and (+1)-stable; (L&beta;/L&beta;+2)&cap;&omega;1=&empty; &#123;zoo 2.18&#125;

S[S&omega;2[&sigma;(&omega; 2) ]] = S[SG[g+&omega;][G[g+&omega;]]] - start 1st 2nd-order gap length &omega; and (+1)-stable; (L&beta;/L&beta;+&omega;)&cap;&omega;1=&empty;

S[S&epsilon; 0 [&sigma;(&epsilon;0)]] = S[SG[g+&epsilon; 0] [G[g+&epsilon;0]]] - start 1st 2nd-order gap length &epsilon;0 and (+1)-stable; (L&beta;/L&beta;+&epsilon; 0 )&cap;&omega;1=&empty;

S[S&alpha;[&sigma;(&alpha;)]] = S[SG[g+&alpha;][G[g+&alpha;]]] - start 1st 2nd-order gap length &alpha;; (L&beta;/L&beta;+&alpha;)&cap;&omega;1=&empty;

S[SG[g&times;2][G[g&times;2]]] = G[g&times;2] - &beta;=(start 1st 2nd-order gap length &beta;); (L&beta;/L&beta;+&beta;)&cap;&omega;1=&empty; &#123;zoo 2.19&#125;

S[SG[gg][G[gg]]] = G[gg] - &beta;=(start 1st 2nd-order gap length &beta;&beta;) &#123;M. Srebrny 1973, Corollary 4.12.&#125;

S[SG[&epsilon; g+1] [G[&epsilon;g+1]]] = G[&epsilon;g+1] - &beta;=(start 1st 2nd-order gap length &beta;&beta;)

G[S[&sigma;](g+1)] - &beta;=(start 1st 2nd-order gap length next admissible after &beta;) [KP+&quot;&omega;1 exists&quot;], &#123;zoo 2.21&#125;

G[S[&sigma;](g+&omega;)] - &beta;=(start 1st 2nd-order gap length next &omega;-th admissible after &beta;) [П21-CA0], [&Delta;22-CA0]

G[S[&sigma;+1](g+1)] - &beta;=(start 1st 2nd-order gap length next (+1)-stable after &beta;)

G[S[S[&sigma;&omega;]](g+1)] - &beta;=(start 1st 2nd-order gap length &omega;-ple stable after &beta;) [П22-CA0], [&Delta;23-CA0]

G[S[S[S2[&sigma;&sigma;+1]]](g+1)] - &beta;=(start 1st 2nd-order gap length next (+1)-2-stable after &beta;)

G[S[S[S2[S[S[S2[&sigma;&sigma;&omega;]]]]]](g+1)] - &beta;=(start 1st 2nd-order gap length &omega;-ple 2-stable after &beta;) [П23-CA0], [&Delta;24-CA0]

G[S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;+1]]]]]]](g+1)] - &beta;=(start 1st 2nd-order gap length next (+1)-3-stable after &beta;)

G[S[S[S2[S[S[S2[S3[S[S[S2[S[S[S2[S3[&sigma;&sigma;&sigma;&omega;]]]]]]]]]]]]](g+1)] - &beta;=(start 1st 2nd-order gap length &omega;-ple 2-stable after &beta;) [П24-CA0], [&Delta;25-CA0]

G[G[g2+1](g+1)] - start 1st 3d-order gap length 1; (L&beta;/L&beta;+1)&cap;&omega;2=&empty; [Z3], [ZFC-+&quot;&omega;1 exists&quot;],  &#123;zoo 2.22&#125;

G[G[S[&sigma;](g2+1)](g+1)] - &beta;=(start 1st 3d-order gap length next admissible after &beta;) [KP+&quot;&omega;2 exists&quot;]

G[G[S[&sigma;](g2+&omega;)](g+1)] - &beta;=(start 1st 3d-order gap length next &omega;-th admissible after &beta;) [П31-CA0], [&Delta;32-CA0]

G[G[S[S[&sigma;&omega;]](g2+1)](g+1)] - &beta;=(start 1st 3d-order gap length &omega;-ple stable after &beta;) [П32-CA0], [&Delta;33-CA0]

G[G[G[g3+1](g2+1)](g+1)] - start 1st 4th-order gap length 1; (L&beta;/L&beta;+1)&cap;&omega;3=&empty; [Z4], [ZFC-+&quot;&omega;2 exists&quot;]

G[G[G[S[&sigma;](g3+1)](g2+1)](g+1)] - &beta;=(start 1st 4th-order gap length next admissible after &beta;) [KP+&quot;&omega;3 exists&quot;]

G[G[G[S[&sigma;](g3+&omega;)](g2+1)](g+1)] - &beta;=(start 1st 4th-order gap length next &omega;-th admissible after &beta;) [П41-CA0], [&Delta;42-CA0]

G[G[G[S[S[&sigma;&omega;]](g3+1)](g2+1)](g+1)] - &beta;=(start 1st 4th-order gap length &omega;-ple stable after &beta;) [П42-CA0], [&Delta;43-CA0]

G[G[G[G[g4+1](g2+1)](g2+1)](g+1)] - start 1st 5th-order gap length 1; (L&beta;/L&beta;+1)&cap;&omega;4=&empty; [Z5], [ZFC-+&quot;&omega;3 exists&quot;]

G[...G[gn+1]...(g+1)] - start 1st n-th-order gap length 1; (L&beta;/L&beta;+1)&cap;&omega;n=&empty; [Zn], [ZFC-+&quot;&omega;n exists&quot;]

G[G[g&omega;]] - [ZFC-+&quot;&omega;&omega; exists&quot;], [Limit of HOA]

G[G[gg]] - [ZFC-+&quot;&omega;(n&lt;&omega; 1) exists&quot;]

G[G[gg 2 ]] - [ZFC-+&quot;&omega;&omega; 1 exists&quot;]

G[G[gg n ]] - [ZFC-+&quot;&omega;&omega; &omega; exists&quot;]

G[G[gg g ]] - [ZFC-+&quot;&omega;&omega; (n&lt;&omega; 1) exists&quot;]

G[G[gg g 2 ]] - [ZFC-+&quot;&omega;&omega; &omega; 1  exists&quot;]

G[G[G2[gg](1undefinedgg0)]] - [ZFC-+&quot;1st fixed point of &omega;&alpha; exists&quot;]

G[G[G2[gg&#39;1]]] - [ZFC-+&quot;admissible and &omega;&alpha; exists&quot;]

G[G[G2[gg&#39;2]]] - [ZFC-+&quot;Mahlo and &omega;&alpha; exists&quot;]

G[G[G2[gg+1]]] - [ZFC-+&quot;(+1)-stable and &omega;&alpha; exists&quot;]

G[G[G2[G[G[G2[gg&omega;]]]]]] - [ZFC-+&quot;&Sigma;2 -correct exists&quot;]

G[G[G2[G[G[G2[G3[G[G[G2[G[G[G2[G3[ggg&omega;]]]]]]]]]]]]] - [ZFC-+&quot;&Sigma;3-correct exists&quot;]

G[G[G2[G[G[G2[G3[G[G[G2[G[G[G2[G3[G4[G[G[G2[G[G[G2[G3[G[G[G2[G[G[G2[G3[G4[gggg&omega;]]]]]]]]]]]]]]]]]]]]]]]]]]]] - [ZFC-+&quot;&Sigma;4-correct exists&quot;]

G[G&omega;[g(&omega;)]] - [ZFC], &#123;zoo 2.24&#125;

 Open question. I still don&#39;t understand what the ordinals &#123;zoo 2.9, 2.14, 2.20, 2.23, 2.25&#125; mean and how to express them.