User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20191006161445/@comment-31580368-20191007020631

Yes, maybe you&#39;re right, and I missed out on these more finer structures. Then my notation should be weaker:

S{&omega;}[&sigma;{&omega;}] = G[g+1] = (1)S((1)&sigma;+1) - start 1st 2nd-order gap length 1; 1st &beta;|(L&beta;/L&beta;+1)&cap;P(&omega;)=&empty;; &beta;|L&beta;⊧Z2; &beta;|L&beta;⊧ZFC-

G{&omega;}[g{&omega;}] = I[i+1] = (2)S((2)&sigma;+1)- start 1st 3d-order gap length 1; 1st &beta;|(L&beta;/L&beta;+1)&cap;P(P(&omega;))=&empty;; &beta;|L&beta;⊧Z3; &beta;|L&beta;⊧ZFC-+&exist;P(&omega;)

I{&omega;}[i{&omega;}] = M[m+1] = (3)S((3)&sigma;+1) - start 1st 4th-order gap length 1; 1st &beta;|(L&beta;/L&beta;+1)&cap;P(P(P(&omega;)))=&empty;; &beta;|L&beta;⊧Z4; &beta;|L&beta;⊧ZFC-+&exist;P(P(&omega;))

e.t.c

(n&lt;&omega;)S((n&lt;&omega;)&sigma;+1) - &beta;|L&beta;⊧Zn; &beta;|L&beta;⊧ZFC-+&forall;n&exist;&omega;n+V=L;

And then it turns out that no extensions like (&alpha;)S((&alpha;)&sigma;+1) have reached &beta;|L&beta;⊧ZFC-+&exist;&omega;&omega; 1 +V=L.

In any case, this does not negate the assumptions that I have suggested here. Because we can introduce the same notation for cardinals &alpha; for use in &beta;|L&beta;⊧ZFC-+&exist;&omega;&alpha;. Something like GSCN (G - Small Cardinal Notation). And if SLCON contains recursive holes, then GSCN will contain recursive and cofinal holes (just as likely we should accept V=L).

G[g] = &omega;

G[g](1) = &omega;1

G[g](1) = &omega;&omega;, 1st П1-reflecting on class P(n)-ordinals

G[g](&alpha;) = &omega;&alpha;

G[g&#39;1] = power-admissible, 1st П2-reflecting on class P(n)-ordinals

G[g&#39;2] = 1st П2-reflecting on П2-reflecting on class P(n)-ordinals

G[g&#39;3] = 1st П3-reflecting on class P(n)-ordinals

G[g+1] - &Sigma;2-correct(&gamma;|V&gamma;≺2Vk,undefined where k - inaccessible cardinal)

G[GG[gg+1]] - &Sigma;3-correct(&gamma;|V&gamma;≺3Vk,undefined where k - inaccessible cardinal)

G[GG[GGG[ggg+1]]] - &Sigma;4-correct(&gamma;|V&gamma;≺4Vk,undefined where k - inaccessible cardinal)

G{&omega;}[g{&omega;}] = (1)G((1)g+1) - gap of cardinality length 1(&gamma;|(V&gamma;/V&gamma;+1)&cap;Vk=&empty;,where k - inaccessible cardinal)

(1)G((1)g+2) - gap of cardinality length 2(&gamma;|(V&gamma;/V&gamma;+2)&cap;Vk=&empty;,where k - inaccessible cardinal)

(1)G{&omega;}[(1)g{&omega;}] = (2)G((2)g+1) - 2nd order gap of cardinality length 1(&gamma;|(V&gamma;/V&gamma;+1)&cap;Vk+1=&empty;,where k - inaccessible cardinal)

(2)G{&omega;}[(2)g{&omega;}] = (3)G((3)g+1) - 3d order gap of cardinality length 1(&gamma;|(V&gamma;/V&gamma;+1)&cap;Vk+1=&empty;,where k - inaccessible cardinal)

(n&lt;&omega;)G((n&lt;&omega;)g+1) - limit of &gamma;|(V&gamma;/V&gamma;+1)&cap;Vk+(n&lt;&omega;)=&empty;(where k - inaccessible cardinal)