## FANDOM

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I'm afraid that Hierarchical nested-subscript array notation isn't as strong as Wythagoras thinks. If I use rules in Bird's Hierarchical hyper-nested array notation, and add a sub-rule, this notation will be not so strong.

This added rule is:

(Rules A5a-b do not apply, separator [Ai(pi)] = [d #1]M, where d ≥ 2 and #1 contains at least one H-hyperseparator in its base layer, where H begins with 1 and H ≠ ‘1’; in other words,

H = ‘1 [H1] 1 [H2] ... 1 [Hk] h #2’, where h ≥ 2, k ≥ 1 and each of [Hi] is a normal separator):

Si = ‘b ‹Ai(1)’› b [Ai(1)] b ‹Ai(2)’› b [Ai(2)] ... b ‹Ai(pi-1)’› b [Ai(pi-1)] Rb [d #1]M ci-1 #i’,

Rn = ‘b ‹Rn-1› b’,

R1 = ‘b [d-1 #1]1 [H1] 1 [H2] ... 1 [Hk-1] 1 ‹Hk’› b (←m+b-1) b’, where m (which may be 1) is the kth and final entry in the subscript array M when written as M = ‘m1 [H1] m2 [H2] ... mk-1 [Hk-1] m’.

Different H-hyperseparators can be placed at the same nested level in HNSAN, so when apply this rule, use the first H-hyperseparator.

In Bird's "separator level" system, a level-$$\alpha$$ separator [A] can handle $$\omega^\alpha$$ entries,and {n,n[A]2} has growth rate $$\omega^{\omega^\alpha}$$ in FGH.

Now I reuse "●" for shorthand of [2/1,22]. and the [1●2] marks the limit of HHNAN. And this table compares their growth rates.

BAN Separator FGH Ordinal
$$[1\bullet2]$$ $$\psi(\Omega_\omega)$$
$$[2\bullet2]$$ $$\psi(\Omega_\omega)^\omega$$
$$[1[1/2]2\bullet2]$$ $$\psi(\Omega_\omega)^{\varepsilon_0}$$
$$[1[1\bullet2]2\bullet2]$$ $$\psi(\Omega_\omega)^{\psi(\Omega_\omega)}$$
$$[1/2\bullet2]$$ $$\psi(\Omega_\omega+1)$$
$$[1/3\bullet2]$$ $$\psi(\Omega_\omega+2)$$
$$[1/1[1/2\bullet2]2\bullet2]$$ $$\psi(\Omega_\omega+\psi(\Omega_\omega+1))$$
$$[1/1/2\bullet2]$$ $$\psi(\Omega_\omega+\Omega)$$
$$[1/2/2\bullet2]$$ $$\psi(\Omega_\omega+\Omega+1)$$
$$[1/1/3\bullet2]$$ $$\psi(\Omega_\omega+\Omega2)$$
$$[1/1/1/2\bullet2]$$ $$\psi(\Omega_\omega+\Omega^2)$$
$$[1[1/2\sim2]2\bullet2]$$ $$\psi(\Omega_\omega+\Omega^\Omega)$$
$$[1[1\sim3]2\bullet2]$$ $$\psi(\Omega_\omega+\varepsilon_{\Omega+1})$$
$$[1[1\sim1\sim2]2\bullet2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_2))$$
$$[1[1[1/_31/_32]2]2\bullet2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_3))$$
$$[1\bullet3]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega))$$
$$[1/2\bullet3]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)+\Omega)$$
$$[1[1\sim1\sim2]2\bullet3]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)+\psi_1(\Omega_2))$$
$$[1\bullet4]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)2)$$
$$[1\bullet1[1\bullet2]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)\psi(\Omega_\omega))$$
$$[1\bullet1/2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)\Omega)$$
$$[1\bullet1[1\sim1\sim2]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)\psi_1(\Omega_2))$$
$$[1\bullet1\bullet2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^2)$$
$$[1\bullet2\bullet2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^2+\psi_1(\Omega_\omega))$$
$$[1\bullet1\bullet3]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^22)$$
$$[1\bullet1\bullet1\bullet2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^3)$$
$$[1[2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^\omega)$$
$$[1\bullet2[2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^\omega+\psi_1(\Omega_\omega))$$
$$[1[2\bullet_22]3]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^\omega2)$$
$$[1[2\bullet_22]1\bullet2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\omega+1})$$
$$[1[2\bullet_22]1[2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\omega2})$$
$$[1[2\bullet_22]1[2\bullet_22]1[2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\omega3})$$
$$[1[3\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\omega^2})$$
$$[1[1[1\bullet2]2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi(\Omega_\omega)})$$
$$[1[1/2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\Omega})$$
$$[1[1\bullet2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)})$$
$$[1[2\bullet2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)\omega})$$
$$[1[1\bullet3\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^2})$$
$$[1[1\bullet1/2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^\Omega})$$
$$[1[1\bullet1\bullet2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)}})$$
$$[1[1\bullet1\bullet3\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)2}})$$
$$[1[1\bullet1\bullet1\bullet2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^2}})$$
$$[1[1[2\bullet_22]2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^\omega}})$$
$$[1[1[1/2\bullet_22]2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^\Omega}})$$
$$[1[1[1\bullet2\bullet_22]2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)}}})$$
$$[1[1[1\bullet1\bullet2\bullet_22]2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)^{\psi_1(\Omega_\omega)}}}})$$
$$[1[1\sim2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega+1))$$
$$[1[1\sim3\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega+2))$$
$$[1[1\sim1/2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega+\Omega))$$
$$[1[1\sim1\bullet2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega+\psi_1(\Omega_\omega)))$$
$$[1[1\sim1[1\sim2\bullet_22]2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_1(\Omega_\omega+\psi_1(\Omega_\omega+1)))$$
$$[1[1\sim1\sim2\bullet_22]2]$$ $$\psi(\Omega_\omega+\Omega_2)$$
$$[1[1[1/_31/_32]2\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_3))$$
$$[1[1\bullet_23]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega))$$
$$[1[1\sim2\bullet_23]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)+\Omega_2)$$
$$[1[1\bullet_24]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)2)$$
$$[1[1\bullet_21\bullet2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)\psi_1(\Omega_\omega))$$
$$[1[1\bullet_21\sim2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)\Omega_2)$$
$$[1[1\bullet_21\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^2)$$
$$[1[1\bullet_22\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^2+\psi_2(\Omega_\omega))$$
$$[1[1\bullet_21\bullet_23]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^22)$$
$$[1[1\bullet_21\bullet_21\bullet_22]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^3)$$
$$[1[1[2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^\omega)$$
$$[1[1[1/2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^\Omega)$$
$$[1[1[1\sim2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^{\Omega_2})$$
$$[1[1[1\bullet_22\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^{\psi_2(\Omega_\omega)})$$
$$[1[1[1\bullet_21\bullet_22\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^{\psi_2(\Omega_\omega)^{\psi_2(\Omega_\omega)}})$$
$$[1[1[1[1\bullet_22\bullet_32]2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega)^{\psi_2(\Omega_\omega)^{\psi_2(\Omega_\omega)^{\psi_2(\Omega_\omega)}}})$$
$$[1[1[1/_32\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega+1))$$
$$[1[1[1/_31/2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega+\Omega))$$
$$[1[1[1/_31\sim2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega+\Omega_2))$$
$$[1[1[1/_31\bullet_22\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega+\psi_2(\Omega_\omega)))$$
$$[1[1[1/_31[1/_32\bullet_32]2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_2(\Omega_\omega+\psi_2(\Omega_\omega+\psi_2(\Omega_\omega))))$$
$$[1[1[1/_31/_32\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\Omega_3)$$
$$[1[1[1[1/_41/_42]2\bullet_32]2]2]$$ $$\psi(\Omega_\omega+\psi_3(\Omega_4))$$
$$[1[1[1\bullet_33]2]2]$$ $$\psi(\Omega_\omega+\psi_3(\Omega_\omega))$$
$$[1[1[1[1/_42\bullet_42]2]2]2]$$ $$\psi(\Omega_\omega+\psi_3(\Omega_\omega+1))$$
$$[1[1[1[1/_41/_42\bullet_42]2]2]2]$$ $$\psi(\Omega_\omega+\Omega_4)$$
$$[1[1[1[1\bullet_43]2]2]2]$$ $$\psi(\Omega_\omega+\psi_4(\Omega_\omega))$$
$$[1[1[1[1[1\bullet_53]2]2]2]2]$$ $$\psi(\Omega_\omega+\psi_5(\Omega_\omega))$$
$$[1[3/_{1,2}2]2]$$ $$\psi(\Omega_\omega2)$$

In Wythagoras's page, he thinks $$\{n,n[1[3/_{1,2}2]2]2\}\approx f_{\psi(\Omega_{\omega2})}(n)$$, which grows much faster than $$f_{\psi(\Omega_\omega2)}(n)$$. That's a key difference.

But, if he uses different rules, it'll be true that $$\{n,n[1[3/_{1,2}2]2]2\}\approx f_{\psi(\Omega_{\omega2})}(n)$$.