User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20191006023842/@comment-31580368-20191006143404

For a more detailed explanation of my assumptions, I will use my notation for LCO.

In this article Hyp cos used stable ordinals as main structures and Пn-reflection ordinals, Пn-reflection on some class ordinals as finer structures.

As a result, he received the following hierarchical structure for analysis, which I can express on my notation:

S[&sigma;] - 1st admissible, &omega;, 1st П1-reflection, &beta;|L&beta;⊧KP

S[&sigma;](1) - 1st admissible after &omega;, 1st П2-reflection, &beta;|L&beta;⊧KP&omega;

S[&sigma;](n&lt;&omega;) - 1st limit of admissible, 1st П1-reflection on П2-reflection, &beta;|L&beta;⊧KPl, &beta;|L&beta;⊧П11-CA0

&alpha;↦S[&sigma;](&alpha;) - 1st fixed point of limit of admissible, 1st П2-reflection, &beta;|L&beta;⊧П11-TR0

S[&sigma;&#39;1] - 1st inaccessible, 1st П2-reflection that is П1-reflection on П2-reflection

S[&sigma;&#39;1](1) - 2nd inaccessible, 2nd П2-reflection that is П1-reflection on П2-reflection

S[&sigma;&#39;2] - 1st Mahlo, 1st П2-reflection on П2-reflection

S[&sigma;&#39;3] - 1st П3-reflection

S[&sigma;&#39;n] - 1st Пn-reflection

S[&sigma;+1] - 1st (+1)-stable, 1st П&omega;-reflecting, 1st &sigma;|L&sigma;≺1L&sigma;+1

S[&sigma;+1&#39;1] - 1st (+1)-П1-reflecting, 1st &sigma;|L&sigma;+1 ⊧ &phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;+1 ⊧ &phi;)|&phi; is П1-formula

S[&sigma;+1] - 1st (+2)-stable, 1st П&omega;&times;2-reflecting, 1st &sigma;|L&sigma;≺1L&sigma;+2

S[&sigma;+&sigma;] - 1st &sigma;=(+&sigma;)-stable, 1st &sigma;|L&sigma;≺1L&sigma;+&sigma;

S[S2[&sigma;2](&sigma;)] - 1st &sigma;=(1st admissible after &sigma;)-stable, 1st &sigma;|L&sigma;≺1L&sigma;+

S[S2[&sigma;2](&sigma;+1)] - 1st &sigma;=(2nd admissible after &sigma;)-stable, 1st &sigma;|L&sigma;≺1L&sigma;++

S[S2[&sigma;2&#39;1](&sigma;)] - 1st &sigma;=(next inaccessible)-stable, 1st inaccessibly-stable ordinal

S[S2[&sigma;2&#39;1](&sigma;+1)] - 2nd &sigma;=(next inaccessible)-stable, 2nd inaccessibly-stable ordinal

S[S2[&sigma;2&#39;2](&sigma;)] - 1st &sigma;=(next Mahlo)-stable, 1st Mahlo-stable ordinal

S[S2[&sigma;2+1](&sigma;)] - 1st doubly (+1)-stable, 1st &sigma;|L&sigma;≺1L&beta;≺1L&beta;+1;

S[S2[S3[&sigma;3+1](&sigma;2)](&sigma;)] - 1st triply (+1)-stable, 1st &sigma;|L&sigma;≺1L&beta;≺1L&gamma;≺1L&gamma;+1

S[Sn&lt;&omega;[&sigma;n&lt;&omega;]] - 1st nonprojectable, 1st П1-reflection on &Sigma;1-stability, &beta;|L&beta;⊧П12-CA0

S[&sigma;&omega;+1] = S[SS[&sigma;&sigma;](&omega;)+1] - 1st &omega;-ply stable, &beta;|L&beta;⊧KP+&Sigma;1-sep, &beta;|L&beta;⊧П12-CA+BI

S[&alpha;↦&sigma;&alpha;] = S[&alpha;↦SS[&sigma;&sigma;](&alpha;)] - 1st fixed point of &alpha;-ply stable, &beta;|L&beta;⊧П12-TR0

S[SS[&sigma;&sigma;&#39;1]] - 1st П2-reflection on &Sigma;1-stability

S[SS[&sigma;&sigma;&#39;2]] - 1st П2-reflection on П2-reflection on &Sigma;1-stability

S[SS[&sigma;&sigma;&#39;3]] - 1st П3-reflection on &Sigma;1-stability

S[SS[&sigma;&sigma;+1]] - 1st (+1)-2-stable, 1st &sigma;|L&sigma;≺2LSt+1, 1st &sigma;=(+1)-stable on (&beta;&lt;&sigma;|L&beta;≺1L&sigma;), 1st &Sigma;2-stable

S[SS[&sigma;&sigma;&omega;+1]] = S[SS[SSS[&sigma;&sigma;&sigma;](&omega;)+1]] - 1st &omega;-ply 2-stable, &beta;|L&beta;⊧KP+&Sigma;2-sep, &beta;|L&beta;⊧П13-CA+BI

S[SS[SSS[&sigma;&sigma;&sigma;&omega;+1]]] = S[SS[SSS[SSSS[&sigma;&sigma;&sigma;&sigma;](&omega;)+1]]] - 1st &omega;-ply 3-stable, &beta;|L&beta;⊧KP+&Sigma;3-sep, &beta;|L&beta;⊧П14-CA+BI

S[S{&omega;}[&sigma;{&omega;}]] - 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-

Finer structures, that Hyp cos used, in my notation is indicated as follows:

&sigma;&#39;1 ~ П2-reflection

&sigma;&#39;2 ~ П2-reflection on П2-reflection

&sigma;&#39;3 ~ П3-reflection

&sigma;+&alpha; = &sigma;&#39;&omega;&times;(&alpha;-1) ~ (+&alpha;)-stable

We can use a similar finer structures for gap-ordinals:

g&#39;1 ~ &Sigma;1 elementary substructure of L

g&#39;2 ~ &Sigma;2 elementary substructure of L

g&#39;3 ~ &Sigma;3 elementary substructure of L

g+&alpha; = g&#39;&omega;&times;(g-1) ~ gap length &alpha;

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

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

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

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

SS2[&sigma;&sigma;2] = S(2)2[&sigma;(2)2] = SG[g&#39;1](1)[G[g&#39;1](1)]

SSS[&sigma;&sigma;&sigma;] = S(3)[&sigma;(3)] = SG[g&#39;2][G[g&#39;2]]

SSS2[&sigma;&sigma;&sigma;2] = S(3)2[&sigma;(3)2] = SG[g&#39;2](1)[G[g&#39;2](1)]

S{&omega;}[&sigma;{&omega;}] = SG[g+1][G[g+1]]

Now the new finer structures is based on &Sigma;n elementary substructure of L and main structures is based on gap-ordinals:

G[g+1] - 1st &beta;|(L&beta;/L&beta;+1)&cap;P(&omega;)=&empty;; &beta;|L&beta;⊧Z2; &beta;|L&beta;⊧ZFC-

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

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

G[S[&sigma;](g)] - 1st &beta;=(start 1st 2nd-order gap length next admissible after &beta;); 1st &beta;|(L&beta;/L&beta;+)&cap;P(&omega;)=&empty;; &beta;|L&beta;⊧KP+&exist;P(&omega;)

G[S[&sigma;](g+1)] - 1st &beta;=(start 1st 2nd-order gap length two next admissible after &beta;); 1st &beta;|(L&beta;/L&beta;++)&cap;P(&omega;)=&empty;

G[S[&sigma;](g+(n&lt;&omega;))] - 1st limit of &beta;=(start 1st 2nd-order gap length n&lt;&omega; admissible after &beta;); &beta;|L&beta;⊧П21-CA0

G[&alpha;↦S[&sigma;](g+&alpha;)] - &beta;|L&beta;⊧П21-TR0

G[S[Sn&lt;&omega;[&sigma;n&lt;&omega;](g)]] -1st limit of &beta;=(start 1st 2nd-order gap length (n&lt;&omega;)-ple stable after &beta;); &beta;|L&beta;⊧П22-CA0

G[G2[g2+1](g)] - 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;)

G[G2[G3[g3+1](g2)](g)] - 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;))

G[Gn&lt;&omega;[gn&lt;&omega;]] - &beta;|L&beta;⊧Zn; &beta;|L&beta;⊧ZFC-+&forall;n&exist;&omega;n+V=L; &beta;|L&beta;⊧ZFC-+&exist;1st П1-reflecting on class P(n)-ordinals

G[g&omega;+1] = G[GG[gg](&omega;)+1] - &beta;|L&beta;⊧ZFC-+&exist;&omega;&omega;+V=L

G[&alpha;↦g&alpha;] = G[&alpha;↦GG[gg](&alpha;)] - &beta;|L&beta;⊧ZFC-+&exist;Beth fixed point

G[GG[gg&#39;1]] - &beta;|L&beta;⊧ZFC-+&exist;power-admissible; &beta;|L&beta;⊧ZFC-+&exist;1st П2-reflecting on class P(n)-ordinals

G[GG[gg+1]] - &beta;|L&beta;⊧ZFC-+&exist;&Sigma;2-extendible(&gamma;|V&gamma;≺2Vk,undefined where k - inaccessible cardinal)

G[GG[GGG[ggg+1]]] - &beta;|L&beta;⊧ZFC-+&exist;&Sigma;3-extendible(&beta;|V&gamma;≺3Vk,undefined where k - inaccessible cardinal)

G{&omega;}[G{&omega;}] - &beta;|L&beta;⊧ZFC; &beta;|L&beta;⊧ZFC-+&exist;&gamma;|(V&gamma;/V&gamma;+1)&cap;Vk=&empty;(where k - inaccessible cardinal); &beta;|L&beta;⊧ZFC-+&exist;least cardinal &beta; that is not definable in ZFC; &beta;|L&beta;⊧ZFC-+&exist;gap of cardinality length 1 exists; &beta;|L&beta;⊧ZFC-+&exist;1st worldly cardinal

We can also introduce new finer structures is based on &Sigma;n-correctness of V and main structures is based on gaps of cardinality:

i&#39;1 ~ &Sigma;1-correctness of V

i&#39;2 ~ &Sigma;2-correctness of V

i&#39;3 ~ &Sigma;3-correctness of V

i+&alpha; = i&#39;&omega;&times;(i-1) ~ gap of cardinality length &alpha;

We need more inductivity:

G[g] = G(1)[g(1)] = GI[i][I[i]]

G2[g2] = G(1)2[g(1)2] = GI[i](1)[I[i](1)]

GG[gg] = G(2)[g(2)] = GI[i&#39;1][I[i&#39;1]]

GG2[gg2] = G(2)2[g(2)2] = GI[i&#39;1](1)[I[i&#39;1](1)]

GGG[ggg] = G(3)[g(3)] = GI[i&#39;2][I[i&#39;2]]

GGG2[ggg2] = G(3)2[g(3)2] = GI[i&#39;2](1)[I[i&#39;2](1)]

G{&omega;}[g{&omega;}] = GI[i+1][I[i+1]]

I[i+1] - &beta;|L&beta;⊧ZFC; &beta;|L&beta;⊧ZFC-+&exist;&gamma;|(V&gamma;/V&gamma;+1)&cap;Vk=&empty;(where k - inaccessible cardinal); &beta;|L&beta;⊧ZFC-+&exist;least cardinal &beta; that is not definable in ZFC; &beta;|L&beta;⊧ZFC-+&exist;gap of cardinality length 1 exists; &beta;|L&beta;⊧ZFC-+&exist;1st worldly cardinal

I[i+2] - &beta;|L&beta;⊧ZFC-+&exist;&gamma;|(V&gamma;/V&gamma;+2)&cap;Vk=&empty;(where k - inaccessible cardinal); &beta;|L&beta;⊧ZFC-+&exist;gap of cardinality length 2 exists; &beta;|L&beta;⊧ZFC-+&exist;2nd worldly cardinal

I[I2[i2+1](i)] - &beta;|L&beta;⊧ZFC-+&exist;&gamma;|(V&gamma;/V&gamma;+1)&cap;Vk+1=&empty;(where k - inaccessible cardinal); &beta;|L&beta;⊧ZFC-+&exist;2nd order gap of cardinality length 1; &beta;|L&beta;⊧ZFC-+&exist;2-worldly cardinal

I[In&lt;&omega;[in&lt;&omega;]] - &beta;|L&beta;⊧ZFC-+&forall;n&exist;&gamma;|(V&gamma;/V&gamma;+1)&cap;Vk+n=&empty;(where k - inaccessible cardinal); &beta;|L&beta;⊧ZFC-+&forall;n&exist;n-worldly cardinal

I[II[ii&#39;1]] - &beta;|L&beta;⊧ZFC-+&exist;hyper-worldly cardinal

I[II[ii+1]] - &beta;|L&beta;⊧ZFC-+&exist;&gamma; that is not &Sigma;1-definable in L using parameters &lt;&gamma;

I[II[III[iii+1]]] - &beta;|L&beta;⊧ZFC-+&exist;&gamma; that is not &Sigma;2-definable in L using parameters &lt;&gamma;

I{&omega;}[I{&omega;}] - &beta;|L&beta;⊧ZFC-+&exist;inaccessible cardinal