FANDOM


Up to \varepsilon_0

IN EA+CHF BAN E^
n | ()() n\uparrow n \{n,n\} En
n | ()()() n\uparrow^2 n \{n,n,2\} En\#n
n | ()()()() n\uparrow^3 n \{n,n,3\} En\#n\#n
n | (()) n\uparrow^\omega n \{n,n,n\} En\#\#n
n | (())() n\uparrow^{\omega+1} n \{n,n,1,2\} En\#\#n\#n
n | (())()() n\uparrow^{\omega+2} n \{n,n,2,2\} En\#\#n\#n\#n
n | (())()() n\uparrow^{\omega+2} n \{n,n,2,2\} En\#\#n\#n\#n
n | (())(()) n\uparrow^{\omega2} n \{n,n,n,2\} En\#\#n\#\#n
n | (())(())(()) n\uparrow^{\omega3} n \{n,n,n,3\} En\#\#n\#\#n\#\#n
n | (())(())(())(()) n\uparrow^{\omega4} n \{n,n,n,4\} En\#\#\#5
n | (()()) n\uparrow^{\omega^2} n \{n,n,n,n\} En\#\#\#n
n | (()())() n\uparrow^{\omega^2+1} n \{n,n,1,1,2\} En\#\#\#n\#n
n | (()())(()) n\uparrow^{\omega^2+\omega} n \{n,n,n,1,2\} En\#\#\#n\#\#n
n | (()())(()()) n\uparrow^{\omega^2 2} n \{n,n,n,n,2\} En\#\#\#n\#\#\#n
n | (()()()) n\uparrow^{\omega^3} n \{n,n,n,n,n\} En\#\#\#\#n
n | (()()()()) n\uparrow^{\omega^4} n \{n,n,n,n,n,n\} En\#\#\#\#\#n
n | ((())) n\uparrow^{\omega^\omega} n \{n,n[2]2\} En#^# n
n | ((()))() n\uparrow^{\omega^\omega+1} n \{n,n,2[2]2\} En#^#n#n
n | ((()))(()) n\uparrow^{\omega^\omega+\omega} n \{n,n,n+1[2]2\} En#^#n##n
n | ((()))((())) n\uparrow^{\omega^\omega2} n \{n,n+2[2]3\} En#^#n#^#n
n | ((()))((()))((())) n\uparrow^{\omega^\omega3} n \{n,n+2[2]4\} En#^#n#^#n#^#n
n | ((())()) n\uparrow^{\omega^{\omega+1}} n \{n,n,...,n[2]n\} En#^#*#n
n | ((())()()) n\uparrow^{\omega^{\omega+2}} n \{n,n,...,n[2]n,n\} En#^#*##n
n | ((())(())) n\uparrow^{\omega^{\omega2}} n \{n,n,...,n[2]n,n,...,n\} En#^#*#^#n
n | ((()())) n\uparrow^{\omega^{\omega^2}} n \{n,n[3]2\} En#^##n
n | ((()()())) n\uparrow^{\omega^{\omega^3}} n \{n,n[4]2\} En#^###n
n | (((()))) n\uparrow^{^3\omega} n  \{n, n [1, 2] 2\} En#^#^#n
n | ((((())))) n\uparrow^{^4\omega} n  \{n, n [1 [2] 2] 2\} En#^#^#^#n
n | (((((()))))) n\uparrow^{^5\omega} n   \{n, n [1 [1, 2] 2] 2\} En#^#^#^#^#n
n |(()_1) n\uparrow^{\varepsilon_0} n    \{n, n [1 / 2] 2\} En#^^#n
n |(()_1) n\uparrow^{\varepsilon_0} n    \{n, n [1 /2] 2\} En#^^#n


Up to \Gamma_0

IN EA+BF FGH BAN xE^
n |(()_1)() n\uparrow^{\psi(\Omega)+1} n f_{\varepsilon_0+1}(n)    \{n,n,2[1/2]2\} En#^^#n#n
n |(()_1()) n\uparrow^{\psi(\Omega+1)} n f_{\varepsilon_0\omega}(n)    \{n,n[1/2]n\} En#^^#*#n
n |(()_1(()_1())) n\uparrow^{\psi(\Omega+\psi(\Omega+1))} n f_{\varepsilon_0^\omega}(n)    \{n,n[2/2]2\} En#^^#^#n
n |(()_1()_1) n\uparrow^{\psi(\Omega2)} n f_{\varepsilon_1}(n)    \{n,n[1/3]2\} En(#^^#)^^#n
n |(()_1()_1()_1) n\uparrow^{\psi(\Omega3)} n f_{\varepsilon_2}(n)    \{n,n[1/4]2\} En((#^^#)^^#)^^#n
n |((())_1) n\uparrow^{\psi(\Omega\omega)} n f_{\varepsilon_\omega}(n)    \{n,n[1/1,2]2\} En#^^#>#n
n |(((()_1))_1) n\uparrow^{\psi(\Omega\psi(\Omega))} n f_{\varepsilon_{\varepsilon_0}}(n)    \{n,n[1/1[1/2]2]2\} En#^^#>#^^#n
n |((()_1)_1) n\uparrow^{\psi(\Omega^2)} n f_{\zeta_0}(n)    \{n,n[1/1/2]2\} En#^^##n
n |((()_1()_1)_1) n\uparrow^{\psi(\Omega^3)} n f_{\varphi(3,0)}(n)    \{n,n[1/1/1/2]2\} En#^^###n
n |((()_1()_1()_1)_1) n\uparrow^{\psi(\Omega^4)} n f_{\varphi(4,0)}(n)    \{n,n[1/1/1/1/2]2\} En#^^#^#4
n |(((())_1)_1) n\uparrow^{\psi(\Omega^\omega)} n f_{\varphi(\omega,0)}(n)    \{n,n[1 [2\sim 2] 2]2\} En#^^#^#n
n |((((()_1))_1)_1) n\uparrow^{\psi(\Omega^{\psi(\Omega)})} n f_{\varphi(\varepsilon_0,0)}(n)    \{n,n[1 [1 [1 / 2] 2 \sim 2] 2]2\} En#^^#^^#n
n |(((()_1)_1)_1) n\uparrow^{\psi(\Omega^\Omega)} n f_{\varphi(1,0,0)}(n)    \{n,n[1 [1 / 2 \sim 2] 2]2\} En#^^^#n
n|(((()_1)_1(()_1)_1)_1) n\uparrow^{\psi(\Omega^{\Omega2})}n f_{\varphi(2,0,0)}(n) \{n,n[1[1/2\sim2]1[1/2\sim2]2]2\} En#{4}#n
n|(((()_1())_1)_1) n\uparrow^{\psi(\Omega^{\Omega\omega})}n f_{\varphi(\omega,0,0)}(n) \{n,n[1[2/2\sim 2]2]2\} En#{n+2}#n


Up to Omega fixed point

IN EA+BF FGH BAN
n |(((()_1()_1)_1)_1) n\uparrow^{\psi(\Omega^{\Omega^2})} n f_{\varphi(1,0,0,0)}(n)    \{n,n[1 [1 / 3 \sim 2] 2]2\}
n |(((()_1()_1()_1)_1)_1) n\uparrow^{\psi(\Omega^{\Omega^3})} n f_{\varphi(1,0,0,0,0)}(n)    \{n,n[1 [1 / 4 \sim 2] 2]2\}
n |(((()_1()_1()_1()_1)_1)_1) n\uparrow^{\psi(\Omega^{\Omega^4})} n f_{\varphi(1,0,0,0,0,0)}(n)    \{n,n[1 [1 / 5 \sim 2] 2]2\}
n |((((())_1)_1)_1) n\uparrow^{\psi(\Omega^{\Omega^\omega})} n f_{\theta(\Omega^\omega)}(n)    \{n,n[1 [1 / 1, 2 \sim 2] 2]2\}
n |((((()_1)_1)_1)_1) n\uparrow^{\psi(\Omega\uparrow\uparrow 3)} n f_{\theta(\Omega^\Omega)}(n)    \{n,n[1 [1 / 1 / 2 \sim 2] 2]2\}
n |(((((()_1)_1)_1)_1)_1) n\uparrow^{\psi(\Omega\uparrow\uparrow 4)} n f_{\theta(\Omega\uparrow\uparrow 3)}(n)    \{n,n[1 [1 [1 / 2 \sim 2] 2 \sim 2] 2]2\}
n |((((((()_1)_1)_1)_1)_1)_1) n\uparrow^{\psi(\Omega\uparrow\uparrow 5)} n f_{\theta(\Omega\uparrow\uparrow 4)}(n)    \{n,n[1 [1 [1 / 1 / 2 \sim 2] 2 \sim 2] 2]2\}
n |(((((((()_1)_1)_1)_1)_1)_1)_1) n\uparrow^{\psi(\Omega\uparrow\uparrow 6)} n f_{\theta(\Omega\uparrow\uparrow 5)}(n)    \{n,n[1 [1 [1 [1 / 2 \sim 2] 2 \sim 2] 2 \sim 2] 2]2\}
n |(()_2) n\uparrow^{\psi(\Omega_2)} n f_{\theta(\varepsilon_{\Omega+1})}(n)    \{n,n[1 [1\sim 3] 2]2\}
n |(((()_2)_2)_2) n\uparrow^{\psi(\Omega_2^{\Omega_2})} n f_{\theta(\Omega_2)}(n)    \{n,n[1 [1 [1 \sim 2 /_3 2] 2] 2]2\}
n |(((()_3)_3)_3) n\uparrow^{\psi(\Omega_3^{\Omega_3})} n f_{\theta(\Omega_3)}(n)    \{n,n[1 [1 [1 [1 /_3 2 /_4 2] 2] 2] 2]2\}
n |(((()_4)_4)_4) n\uparrow^{\psi(\Omega_4^{\Omega_4})} n f_{\theta(\Omega_4)}(n)    \{n,n[1 [1 [1 [1 [1 /_4 2 /_5 2] 2] 2] 2] 2]2\}
n\uparrow^{\psi(\Lambda)} n f_{\psi(\psi_\Iota(0))}(n)

Note:

IN denotes I-notation;

EA+CHF denotes Extended Arrows with ordinals \alpha\le\varepsilon_0 written in Cantor normal form;

EA+BF denotes Extended Arrows with ordinals \alpha\le\Lambda written in normal form for Buchholz function whose fundamental sequences are assigned according ruleset for this function;

BAN denotes Bird's array notation;

E^ denotes Cascading-E notation;

xE^ denotes Extended Cascading-E Notation;


Appendix: Extended Arrows's Definition

We define for non-zero natural numbers n, b and for ordinal number \alpha\geq 0:

1) n\uparrow^\alpha b= nb \text{ if }\alpha=0,

2) n\uparrow^{\alpha+1} b = \left\{\begin{array}{lcr} n \text{ if }b=1\\ n\uparrow^{\alpha}(n\uparrow^{\alpha+1} (b-1))\text{ if }b>1\\ \end{array}\right.,

3) n\uparrow^{\alpha} b=n\uparrow^{\alpha[b]} n iff \alpha is a limit ordinal,

where \alpha [b] denotes the b-th element of the fundamental sequence assigned to the limit ordinal \alpha.

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