FANDOM


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 \(\{n,n[1[3/_{1,2}2]2]2\}\approx f_{\psi(\Omega_{\omega2})}(n)\).

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