UPDATE March 6th, 2026: I diverged the systems of the x&E into two systems: one that maps to the limit of extended Buchholz's function, and this one that take account more effectively to the Bashicu matrix system (BMS) and Y sequence. Instead of going straight to &_2, we have to define &^&^&^...^&^&^& = &^^# = &(1), &(1)^^# = &(2), &(2)^^# = &(3), etc. based off the unofficial extension of Madore's OCF that the limit of &(&(&(...&(&(&(1)))...))) is &(&_2). Similarly, after &(&_2^&_2^&_2^...), we have &(&_2(1)), &(&_2(&_3)), &(&_2(&_3(1))), &(&_2(&_3(&_4(1)))), and so on, until we reach the limit at #{/}# = #{/(1)}# = #{&(&_2(&_3(&_4(...&_{n-2}(&_{n-1}(&_n(1)))...))))}#, which has the Buchholz's ordinal level (ψ_0(Ω_ω) using Buchholz's OCF), and is also equal to (0,0,0)(1,1,1) in Bashicu matrix system (BMS) and (1,2,4,8) in Y sequence.
First of all, we observe the pattern:
#{&^^#}# = #{&^&^&^&^&^&^&^&^...}#
#{&^^#}#[n] = #{&^&^&^&^...^&^&^&^&}# with n &'s
So, #{&^^#}# = #{&_2}# in x&E1
But, in the second system of extended Collapsing-E notation (x&E2), we have:
#{&^^#}# = #{&(1)}#, where &(0) = &
So:
#{&_2}# in x&E1 = #{&(1)}# in x&E2
And "&" is the shorthand for "&_1".
Before defining the formal rules of the extended Collapsing-E notation system 2, we should have the analysis of the intended growth rates as follows:
#{&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0))
Buchholz: ψ_0(Ω_2)
BMS: (0,0)(1,1)(2,2)
Y sequence: (1,2,4,7)
(#{&(1)}#)^^# has ordinal level:
Madore: ψ(ψ_Ω_2(0) + 1)
Buchholz: ψ_0(Ω_2 + Ω)
BMS: (0,0)(1,1)(2,2)(1,1)
Y sequence: (1,2,4,7,4)
(#{&(1)}#)^^#># has ordinal level:
Madore: ψ(ψ_Ω_2(0) + ω)
Buchholz: ψ_0(Ω_2 + Ω·ω)
BMS: (0,0)(1,1)(2,2)(1,1)(2,0)
Y sequence: (1,2,4,7,4,5)
(#{&(1)}#)^^## has ordinal level:
Madore: ψ(ψ_Ω_2(0) + Ω)
Buchholz: ψ_0(Ω_2 + Ω^2)
BMS: (0,0)(1,1)(2,2)(1,1)(2,1)
Y sequence: (1,2,4,7,4,6)
(#{&(1)}#)^^^# has ordinal level:
Madore: ψ(ψ_Ω_2(0) + Ω^Ω)
Buchholz: ψ_0(Ω_2 + Ω^Ω)
BMS: (0,0)(1,1)(2,2)(1,1)(2,1)(3,1)
Y sequence: (1,2,4,7,4,6,8)
(#{&(1)}#){&^&}# has ordinal level:
Madore: ψ(ψ_Ω_2(0) + Ω^Ω^Ω)
Buchholz: ψ_0(Ω_2 + Ω^Ω^Ω)
BMS: (0,0)(1,1)(2,2)(1,1)(2,1)(3,1)(4,1)
Y sequence: (1,2,4,7,4,6,8,10)
(#{&(1)}#){&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·2)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2))
BMS: (0,0)(1,1)(2,2)(1,1)(2,2)
Y sequence: (1,2,4,7,4,7)
((#{&(1)}#){&(1)}#){&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·3)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2)·2)
BMS: (0,0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,2)
Y sequence: (1,2,4,7,4,7,4,7)
#{&(1)}#># has ordinal level:
Madore: ψ(ψ_Ω_2(0)·ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + 1))
BMS: (0,0)(1,1)(2,2)(2,0)
Y sequence: (1,2,4,7,5)
#{&(1)}#>#{&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·ψ_Ω_2(0))
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_0(Ω_2)))
BMS: (0,0)(1,1)(2,2)(2,0)(3,1)(4,2)
Y sequence: (1,2,4,7,5,7,10)
#{&(1)}## has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω))
BMS: (0,0)(1,1)(2,2)(2,1)
Y sequence: (1,2,4,7,6)
#{&(1)}### has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω^2)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω·2))
BMS: (0,0)(1,1)(2,2)(2,1)(2,1)
Y sequence: (1,2,4,7,6,6)
#{&(1)}#^# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω^ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω·ω))
BMS: (0,0)(1,1)(2,2)(2,1)(3,0)
Y sequence: (1,2,4,7,6,7)
#{&(1)+1}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω^Ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω^2))
BMS: (0,0)(1,1)(2,2)(2,1)(3,1)
Y sequence: (1,2,4,7,6,8)
#{&(1)+&}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω^Ω^2)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω^3))
BMS: (0,0)(1,1)(2,2)(2,1)(3,1)(3,1)
Y sequence: (1,2,4,7,6,8,8)
#{&(1)+&^#}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω^Ω^ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω^ω))
BMS: (0,0)(1,1)(2,2)(2,1)(3,1)(4,0)
Y sequence: (1,2,4,7,6,8,9)
#{&(1)+&^&}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω^Ω^Ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω^Ω))
BMS: (0,0)(1,1)(2,2)(2,1)(3,1)(4,1)
Y sequence: (1,2,4,7,6,8,10)
#{&(1)+&^&^&}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)·Ω^Ω^Ω^Ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + Ω^Ω^Ω))
BMS: (0,0)(1,1)(2,2)(2,1)(3,1)(4,1)(5,1)
Y sequence: (1,2,4,7,6,8,10,12)
#{&(1)+&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^2)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2)))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)
Y sequence: (1,2,4,7,6,9)
#{&(1)+&(1)+&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^3)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2)·2))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(2,1)(3,2)
Y sequence: (1,2,4,7,6,9,6,9)
#{&(1)*#}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + 1)))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,0)
Y sequence: (1,2,4,7,6,9,7)
#{&(1)*&}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^Ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + Ω)))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)
Y sequence: (1,2,4,7,6,9,8)
#{&(1)*&^&}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^Ω^Ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + Ω^2)))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,1)
Y sequence: (1,2,4,7,6,9,8,10)
#{&(1)*&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^ψ_Ω_2(0))
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2))))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)
Y sequence: (1,2,4,7,6,9,8,11)
#{&(1)*&(1)*&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^ψ_Ω_2(0)^2)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2)·2)))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)(3,1)(4,2)
Y sequence: (1,2,4,7,6,9,8,11,8,11)
#{&(1)^#}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^ψ_Ω_2(0)^ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + 1))))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)(4,0)
Y sequence: (1,2,4,7,6,9,8,11,9)
#{&(1)^&}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^ψ_Ω_2(0)^Ω)
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + Ω))))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)(4,1)
Y sequence: (1,2,4,7,6,9,8,11,10)
#{&(1)^&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^ψ_Ω_2(0)^ψ_Ω_2(0))
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2)))))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)(4,1)(5,2)
Y sequence: (1,2,4,7,6,9,8,11,10,13)
#{&(1)^&(1)^&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(0)^ψ_Ω_2(0)^ψ_Ω_2(0)^ψ_Ω_2(0))
Buchholz: ψ_0(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2 + ψ_1(Ω_2))))))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)(4,1)(5,2)(5,1)(6,2)
Y sequence: (1,2,4,7,6,9,8,11,10,13,12,15)
#{&(2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(1))
Buchholz: ψ_0(Ω_2·2)
BMS: (0,0)(1,1)(2,2)(2,2)
Y sequence: (1,2,4,7,7)
(#{&(2)}#){&(2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(1)·2)
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2))
BMS: (0,0)(1,1)(2,2)(2,2)(1,1)(2,2)(2,2)
Y sequence: (1,2,4,7,7,4,7,7)
#{&(2)}#># has ordinal level:
Madore: ψ(ψ_Ω_2(1)·ω)
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + 1))
BMS: (0,0)(1,1)(2,2)(2,2)(2,0)
Y sequence: (1,2,4,7,7,5)
#{&(2)}## has ordinal level:
Madore: ψ(ψ_Ω_2(1)·Ω)
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + Ω))
BMS: (0,0)(1,1)(2,2)(2,2)(2,1)
Y sequence: (1,2,4,7,7,6)
#{&(2)+&(1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(1)·ψ_Ω_2(0))
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2)))
BMS: (0,0)(1,1)(2,2)(2,2)(2,1)(3,2)
Y sequence: (1,2,4,7,7,6,9)
#{&(2)+&(2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(1)^2)
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2)))
BMS: (0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2)
Y sequence: (1,2,4,7,7,6,9,9)
#{&(2)*#}# has ordinal level:
Madore: ψ(ψ_Ω_2(1)^ω)
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2 + 1)))
BMS: (0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2)(3,0)
Y sequence: (1,2,4,7,7,6,9,9,7)
#{&(2)*&}# has ordinal level:
Madore: ψ(ψ_Ω_2(1)^Ω)
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2 + Ω)))
BMS: (0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2)(3,1)
Y sequence: (1,2,4,7,7,6,9,9,8)
#{&(2)*&(2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(1)^ψ_Ω_2(1))
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2))))
BMS: (0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2)(3,1)(4,2)(4,2)
Y sequence: (1,2,4,7,7,6,9,9,8,11,11)
#{&(2)^&(2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(1)^ψ_Ω_2(1)^ψ_Ω_2(1))
Buchholz: ψ_0(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2 + ψ_1(Ω_2·2)))))
BMS: (0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2)(3,1)(4,2)(4,2)(4,1)(5,2)(5,2)
Y sequence: (1,2,4,7,7,6,9,9,8,11,11,10,13,13)
#{&(3)}# has ordinal level:
Madore: ψ(ψ_Ω_2(2))
Buchholz: ψ_0(Ω_2·3)
BMS: (0,0)(1,1)(2,2)(2,2)(2,2)
Y sequence: (1,2,4,7,7,7)
#{&(3)+&(3)}# has ordinal level:
Madore: ψ(ψ_Ω_2(2)^2)
Buchholz: ψ_0(Ω_2·3 + ψ_1(Ω_2·3 + ψ_1(Ω_2·3)))
BMS: (0,0)(1,1)(2,2)(2,2)(2,2)(2,1)(3,2)(3,2)(3,2)
Y sequence: (1,2,4,7,7,7,6,9,9,9)
#{&(3)*&(3)}# has ordinal level:
Madore: ψ(ψ_Ω_2(2)^ψ_Ω_2(2))
Buchholz: ψ_0(Ω_2·3 + ψ_1(Ω_2·3 + ψ_1(Ω_2·3 + ψ_1(Ω_2·3))))
BMS: (0,0)(1,1)(2,2)(2,2)(2,2)(2,1)(3,2)(3,2)(3,2)(3,1)(4,2)(4,2)(4,2)
Y sequence: (1,2,4,7,7,7,6,9,9,9,8,11,11,11)
#{&(4)}# has ordinal level:
Madore: ψ(ψ_Ω_2(3))
Buchholz: ψ_0(Ω_2·4)
BMS: (0,0)(1,1)(2,2)(2,2)(2,2)(2,2)
Y sequence: (1,2,4,7,7,7,7)
#{&(5)}# has ordinal level:
Madore: ψ(ψ_Ω_2(4))
Buchholz: ψ_0(Ω_2·5)
BMS: (0,0)(1,1)(2,2)(2,2)(2,2)(2,2)(2,2)
Y sequence: (1,2,4,7,7,7,7,7)
#{&(#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ω))
Buchholz: ψ_0(Ω_2·ω)
BMS: (0,0)(1,1)(2,2)(3,0)
Y sequence: (1,2,4,7,8)
#{&(#+1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ω + 1))
Buchholz: ψ_0(Ω_2·(ω + 1))
BMS: (0,0)(1,1)(2,2)(3,0)(2,2)
Y sequence: (1,2,4,7,8,7)
#{&(#+#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ω·2))
Buchholz: ψ_0(Ω_2·ω·2)
BMS: (0,0)(1,1)(2,2)(3,0)(2,2)(3,0)
Y sequence: (1,2,4,7,8,7,8)
#{&(##)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ω^2))
Buchholz: ψ_0(Ω_2·ω^2)
BMS: (0,0)(1,1)(2,2)(3,0)(3,0)
Y sequence: (1,2,4,7,8,8)
#{&(#^#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ω^ω))
Buchholz: ψ_0(Ω_2·ω^ω)
BMS: (0,0)(1,1)(2,2)(3,0)(4,0)
Y sequence: (1,2,4,7,8,9)
#{&(#^^#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ(0)))
Buchholz: ψ_0(Ω_2·ψ_0(Ω))
BMS: (0,0)(1,1)(2,2)(3,0)(4,1)
Y sequence: (1,2,4,7,8,10)
#{&(#{&(1)}#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ(ψ_Ω_2(0))))
Buchholz: ψ_0(Ω_2·ψ_0(Ω_2))
BMS: (0,0)(1,1)(2,2)(3,0)(4,1)(5,2)
Y sequence: (1,2,4,7,8,10,13)
#{&(&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω))
Buchholz: ψ_0(Ω_2·Ω)
BMS: (0,0)(1,1)(2,2)(3,1)
Y sequence: (1,2,4,7,9)
#{&(&+1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω + 1))
Buchholz: ψ_0(Ω_2·(Ω + 1))
BMS: (0,0)(1,1)(2,2)(3,1)(2,2)
Y sequence: (1,2,4,7,9,7)
#{&(&+&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω·2))
Buchholz: ψ_0(Ω_2·Ω·2)
BMS: (0,0)(1,1)(2,2)(3,1)(2,2)(3,1)
Y sequence: (1,2,4,7,9,7,9)
#{&(&#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω·ω))
Buchholz: ψ_0(Ω_2·Ω·ω)
BMS: (0,0)(1,1)(2,2)(3,1)(3,0)
Y sequence: (1,2,4,7,9,8)
#{&(&&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω^2))
Buchholz: ψ_0(Ω_2·Ω^2)
BMS: (0,0)(1,1)(2,2)(3,1)(3,1)
Y sequence: (1,2,4,7,9,9)
#{&(&&&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω^3))
Buchholz: ψ_0(Ω_2·Ω^3)
BMS: (0,0)(1,1)(2,2)(3,1)(3,1)(3,1)
Y sequence: (1,2,4,7,9,9,9)
#{&(&^#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω^ω))
Buchholz: ψ_0(Ω_2·Ω^ω)
BMS: (0,0)(1,1)(2,2)(3,1)(4,0)
Y sequence: (1,2,4,7,9,10)
#{&(&^&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω^Ω))
Buchholz: ψ_0(Ω_2·Ω^Ω)
BMS: (0,0)(1,1)(2,2)(3,1)(4,1)
Y sequence: (1,2,4,7,9,11)
#{&(&^&^&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω^Ω^Ω))
Buchholz: ψ_0(Ω_2·Ω^Ω^Ω)
BMS: (0,0)(1,1)(2,2)(3,1)(4,1)(5,1)
Y sequence: (1,2,4,7,9,11,13)
#{&(&(1))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_2(0)))
Buchholz: ψ_0(Ω_2·ψ_1(Ω_2))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)
Y sequence: (1,2,4,7,9,12)
#{&(&(2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_2(1)))
Buchholz: ψ_0(Ω_2·ψ_1(Ω_2·2))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)(4,2)
Y sequence: (1,2,4,7,9,12,12)
#{&(&(#))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_2(1)))
Buchholz: ψ_0(Ω_2·ψ_1(Ω_2·ω))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)(5,0)
Y sequence: (1,2,4,7,9,12,13)
#{&(&(&))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_2(Ω)))
Buchholz: ψ_0(Ω_2·ψ_1(Ω_2·Ω))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)(5,1)
Y sequence: (1,2,4,7,9,12,14)
#{&(&(&(1)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_2(ψ_Ω_2(0))))
Buchholz: ψ_0(Ω_2·ψ_1(Ω_2·ψ_1(Ω_2)))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,2)
Y sequence: (1,2,4,7,9,12,14,17)
#{&(&(&(&(1))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_2(ψ_Ω_2(ψ_Ω_2(0)))))
Buchholz: ψ_0(Ω_2·ψ_1(Ω_2·ψ_1(Ω_2·ψ_1(Ω_2))))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,2)(7,1)(8,2)
Y sequence: (1,2,4,7,9,12,14,17,19,22)
And now, let's introduce the &_2 hyper-delimiter!
#{&(&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2))
Buchholz: ψ_0(Ω_2^2)
BMS: (0,0)(1,1)(2,2)(3,2)
Y sequence: (1,2,4,7,10)
#{&(&_2+1)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2 + 1))
Buchholz: ψ_0(Ω_2^2 + Ω_2)
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)
Y sequence: (1,2,4,7,10,7)
#{&(&_2+2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2 + 2))
Buchholz: ψ_0(Ω_2^2 + Ω_2·2)
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)(2,2)
Y sequence: (1,2,4,7,10,7,7)
#{&(&_2+#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2 + ω))
Buchholz: ψ_0(Ω_2^2 + Ω_2·ω)
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)(3,0)
Y sequence: (1,2,4,7,10,7,8)
#{&(&_2+&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2 + Ω))
Buchholz: ψ_0(Ω_2^2 + Ω_2·Ω)
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)(3,1)
Y sequence: (1,2,4,7,10,7,9)
#{&(&_2+&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2·2))
Buchholz: ψ_0(Ω_2^2·2)
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)(3,2)
Y sequence: (1,2,4,7,10,7,10)
#{&(&_2*#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2·ω))
Buchholz: ψ_0(Ω_2^2·ω)
BMS: (0,0)(1,1)(2,2)(3,2)(3,0)
Y sequence: (1,2,4,7,10,8)
#{&(&_2*&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2·Ω))
Buchholz: ψ_0(Ω_2^2·Ω)
BMS: (0,0)(1,1)(2,2)(3,2)(3,1)
Y sequence: (1,2,4,7,10,9)
#{&(&_2*&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^2))
Buchholz: ψ_0(Ω_2^3)
BMS: (0,0)(1,1)(2,2)(3,2)(3,2)
Y sequence: (1,2,4,7,10,10)
#{&(&_2*&_2*&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^3))
Buchholz: ψ_0(Ω_2^4)
BMS: (0,0)(1,1)(2,2)(3,2)(3,2)(3,2)
Y sequence: (1,2,4,7,10,10,10)
#{&(&_2^#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^ω))
Buchholz: ψ_0(Ω_2^ω)
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)
Y sequence: (1,2,4,7,10,11)
#{&(&_2^&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω))
Buchholz: ψ_0(Ω_2^Ω)
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)
Y sequence: (1,2,4,7,10,12)
#{&(&_2^&^&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω^Ω))
Buchholz: ψ_0(Ω_2^Ω^Ω)
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,1)
Y sequence: (1,2,4,7,10,12,14)
#{&(&_2^&(1))}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^ψ_Ω_2(0)))
Buchholz: ψ_0(Ω_2^ψ_1(Ω_2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,2)
Y sequence: (1,2,4,7,10,12,15)
#{&(&_2^&(&_2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^ψ_Ω_2(Ω_2)))
Buchholz: ψ_0(Ω_2^ψ_1(Ω_2^2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)
Y sequence: (1,2,4,7,10,12,15,18)
#{&(&_2^&(&_2^&(&_2)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^ψ_Ω_2(Ω_2^ψ_Ω_2(Ω_2))))
Buchholz: ψ_0(Ω_2^ψ_1(Ω_2^ψ_1(Ω_2^2)))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)(8,2)(9,2)
Y sequence: (1,2,4,7,10,12,15,18,20,23,26)
#{&(&_2^&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2))
Buchholz: ψ_0(Ω_2^Ω_2)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)
Y sequence: (1,2,4,7,10,13)
#{&(&_2^&_2*&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^(Ω_2 + 1)))
Buchholz: ψ_0(Ω_2^(Ω_2 + 1))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(3,2)
Y sequence: (1,2,4,7,10,13,10)
#{&(&_2^&_2*&_2^&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^(Ω_2·2)))
Buchholz: ψ_0(Ω_2^(Ω_2·2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(3,2)(4,2)
Y sequence: (1,2,4,7,10,13,10,13)
#{&(&_2^(&_2*#))}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^(Ω_2·ω)))
Buchholz: ψ_0(Ω_2^(Ω_2·ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(4,0)
Y sequence: (1,2,4,7,10,13,11)
#{&(&_2^(&_2*&))}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^(Ω_2·Ω)))
Buchholz: ψ_0(Ω_2^(Ω_2·Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(4,1)
Y sequence: (1,2,4,7,10,13,12)
#{&(&_2^(&_2*&_2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^2))
Buchholz: ψ_0(Ω_2^Ω_2^2)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(4,2)
Y sequence: (1,2,4,7,10,13,13)
#{&(&_2^&_2^#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^ω))
Buchholz: ψ_0(Ω_2^Ω_2^ω)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,0)
Y sequence: (1,2,4,7,10,13,14)
#{&(&_2^&_2^&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^Ω))
Buchholz: ψ_0(Ω_2^Ω_2^Ω)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,1)
Y sequence: (1,2,4,7,10,13,15)
#{&(&_2^&_2^&(&_2^&_2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^ψ_Ω_2(Ω_2^Ω_2)))
Buchholz: ψ_0(Ω_2^Ω_2^ψ_1(Ω_2^Ω_2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,1)(6,2)(7,2)(8,2)
Y sequence: (1,2,4,7,10,13,15,18,21,24)
#{&(&_2^&_2^&_2}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^Ω_2))
Buchholz: ψ_0(Ω_2^Ω_2^Ω_2)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)
Y sequence: (1,2,4,7,10,13,16)
#{&(&_2^&_2^&_2^&}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^Ω_2^Ω))
Buchholz: ψ_0(Ω_2^Ω_2^Ω_2^Ω)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,1)
Y sequence: (1,2,4,7,10,13,16,18)
#{&(&_2^&_2^&_2^&_2}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^Ω_2^Ω_2))
Buchholz: ψ_0(Ω_2^Ω_2^Ω_2^Ω_2)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2)
Y sequence: (1,2,4,7,10,13,16,19)
#{&(&_2^&_2^&_2^&_2^&_2}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2))
Buchholz: ψ_0(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2)(7,2)
Y sequence: (1,2,4,7,10,13,16,19,22)
#{&(&_2^&_2^&_2^&_2^&_2^&_2}# has ordinal level:
Madore: ψ(ψ_Ω_2(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2))
Buchholz: ψ_0(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2)(7,2)(8,2)
Y sequence: (1,2,4,7,10,13,16,19,22,25)
Let's move on the omega-3 uncountable level!
#{&(&_2(1))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(0)))
Buchholz: ψ_0(Ω_3)
BMS: (0,0)(1,1)(2,2)(3,3)
Y sequence: (1,2,4,7,11)
#{&(&_2(1)+&_2(1))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(0)·2))
Buchholz: ψ_0(Ω_3 + ψ_2(Ω_3))
BMS: (0,0)(1,1)(2,2)(3,3)(2,2)(3,3)
Y sequence: (1,2,4,7,11,7,11)
#{&(&_2(1)*#)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(0)·ω))
Buchholz: ψ_0(Ω_3 + ψ_2(Ω_3 + 1))
BMS: (0,0)(1,1)(2,2)(3,3)(3,0)
Y sequence: (1,2,4,7,11,8)
#{&(&_2(1)*&)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(0)·Ω))
Buchholz: ψ_0(Ω_3 + ψ_2(Ω_3 + Ω))
BMS: (0,0)(1,1)(2,2)(3,3)(3,1)
Y sequence: (1,2,4,7,11,9)
#{&(&_2(1)*&_2)}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(0)·Ω_2))
Buchholz: ψ_0(Ω_3 + ψ_2(Ω_3 + Ω_2))
BMS: (0,0)(1,1)(2,2)(3,3)(3,2)
Y sequence: (1,2,4,7,11,10)
#{&(&_2(1)*&_2(1))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(0)^2))
Buchholz: ψ_0(Ω_3 + ψ_2(Ω_3 + ψ_2(Ω_3)))
BMS: (0,0)(1,1)(2,2)(3,3)(3,2)(4,3)
Y sequence: (1,2,4,7,11,10,14)
#{&(&_2(1)^&_2(1))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(0)^ψ_Ω_3(0)))
Buchholz: ψ_0(Ω_3 + ψ_2(Ω_3 + ψ_2(Ω_3 + ψ_2(Ω_3))))
BMS: (0,0)(1,1)(2,2)(3,3)(3,2)(4,3)(4,2)(5,3)
Y sequence: (1,2,4,7,11,10,14,13,18)
#{&(&_2(2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(1)))
Buchholz: ψ_0(Ω_3·2)
BMS: (0,0)(1,1)(2,2)(3,3)(3,3)
Y sequence: (1,2,4,7,11,11)
#{&(&_2(3))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(2)))
Buchholz: ψ_0(Ω_3·3)
BMS: (0,0)(1,1)(2,2)(3,3)(3,3)(3,3)
Y sequence: (1,2,4,7,11,11,11)
#{&(&_2(#))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ω)))
Buchholz: ψ_0(Ω_3·ω)
BMS: (0,0)(1,1)(2,2)(3,3)(4,0)
Y sequence: (1,2,4,7,11,12)
#{&(&_2(&))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω)))
Buchholz: ψ_0(Ω_3·Ω)
BMS: (0,0)(1,1)(2,2)(3,3)(4,1)
Y sequence: (1,2,4,7,11,13)
#{&(&_2(&(1)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_2(0))))
Buchholz: ψ_0(Ω_3·ψ_1(Ω_2))
BMS: (0,0)(1,1)(2,2)(3,3)(4,1)(5,2)
Y sequence: (1,2,4,7,11,13,16)
#{&(&_2(&(&_2(0))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_2(ψ_Ω_3(0)))))
Buchholz: ψ_0(Ω_3·ψ_1(Ω_3))
BMS: (0,0)(1,1)(2,2)(3,3)(4,1)(5,2)(6,3)
Y sequence: (1,2,4,7,11,13,16,20)
#{&(&_2(&_2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_2)))
Buchholz: ψ_0(Ω_3·Ω_2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,2)
Y sequence: (1,2,4,7,11,14)
#{&(&_2(&_2(1)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_3(0))))
Buchholz: ψ_0(Ω_3·ψ_2(Ω_3))
BMS: (0,0)(1,1)(2,2)(3,3)(4,2)(5,3)
Y sequence: (1,2,4,7,11,14,18)
#{&(&_2(&_2(&_2(1))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_3(ψ_Ω_3(0)))))
Buchholz: ψ_0(Ω_3·ψ_2(Ω_3·ψ_2(Ω_3)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,2)(5,3)(6,2)(7,3)
Y sequence: (1,2,4,7,11,14,18,21,25)
#{&(&_2(&_3))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3)))
Buchholz: ψ_0(Ω_3^2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)
Y sequence: (1,2,4,7,11,15)
#{&(&_2(&_3*&_3))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^2)))
Buchholz: ψ_0(Ω_3^3)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(4,3)
Y sequence: (1,2,4,7,11,15,15)
#{&(&_2(&_3^#))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^ω)))
Buchholz: ψ_0(Ω_3^ω)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,0)
Y sequence: (1,2,4,7,11,15,16)
#{&(&_2(&_3^&))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^Ω)))
Buchholz: ψ_0(Ω_3^Ω)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,1)
Y sequence: (1,2,4,7,11,15,17)
#{&(&_2(&_3^&_2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^Ω_2)))
Buchholz: ψ_0(Ω_3^Ω_2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,2)
Y sequence: (1,2,4,7,11,15,18)
#{&(&_2(&_3^&_2(&_3)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^ψ_Ω_3(Ω_3))))
Buchholz: ψ_0(Ω_3^ψ_2(Ω_3))
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,2)(6,3)(7,3)
Y sequence: (1,2,4,7,11,15,18,22,26)
#{&(&_2(&_3^&_3))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^Ω_3)))
Buchholz: ψ_0(Ω_3^Ω_3)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,3)
Y sequence: (1,2,4,7,11,15,19)
#{&(&_2(&_3^&_3^&_2))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^Ω_3^Ω_2)))
Buchholz: ψ_0(Ω_3^Ω_3^Ω_2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,3)(6,2)
Y sequence: (1,2,4,7,11,15,19,22)
#{&(&_2(&_3^&_3^&_3))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^Ω_3^Ω_3)))
Buchholz: ψ_0(Ω_3^Ω_3^Ω_3)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,3)(6,3)
Y sequence: (1,2,4,7,11,15,19,23)
#{&(&_2(&_3^&_3^&_3^&_3))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(Ω_3^Ω_3^Ω_3^Ω_3)))
Buchholz: ψ_0(Ω_3^Ω_3^Ω_3^Ω_3)
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,3)(6,3)(7,3)
Y sequence: (1,2,4,7,11,15,19,23,27)
And let's run to the end!
#{&(&_2(&_3(1)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(0))))
Buchholz: ψ_0(Ω_4)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)
Y sequence: (1,2,4,7,11,16)
#{&(&_2(&_3(2)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(1))))
Buchholz: ψ_0(Ω_4·2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(4,4)
Y sequence: (1,2,4,7,11,16,16)
#{&(&_2(&_3(#)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ω))))
Buchholz: ψ_0(Ω_4·ω)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,0)
Y sequence: (1,2,4,7,11,16,17)
#{&(&_2(&_3(&)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(Ω))))
Buchholz: ψ_0(Ω_4·Ω)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,1)
Y sequence: (1,2,4,7,11,16,18)
#{&(&_2(&_3(&_2)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(Ω_2))))
Buchholz: ψ_0(Ω_4·Ω_2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,2)
Y sequence: (1,2,4,7,11,16,19)
#{&(&_2(&_3(&_3)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(Ω_3))))
Buchholz: ψ_0(Ω_4·Ω_3)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,3)
Y sequence: (1,2,4,7,11,16,20)
#{&(&_2(&_3(&_3(1))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_4(0)))))
Buchholz: ψ_0(Ω_4·ψ_3(Ω_4))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,3)(6,4)
Y sequence: (1,2,4,7,11,16,20,25)
#{&(&_2(&_3(&_4)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(Ω_4))))
Buchholz: ψ_0(Ω_4^2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,4)
Y sequence: (1,2,4,7,11,16,21)
#{&(&_2(&_3(&_4^&_3)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(Ω_4^Ω_3))))
Buchholz: ψ_0(Ω_4^Ω_4)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,4)(6,3)
Y sequence: (1,2,4,7,11,16,21,25)
#{&(&_2(&_3(&_4^&_4)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(Ω_4^Ω_4))))
Buchholz: ψ_0(Ω_4^Ω_4)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,4)(6,4)
Y sequence: (1,2,4,7,11,16,21,26)
#{&(&_2(&_3(&_4^&_4^&_4)))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(Ω_4^Ω_4^Ω_4))))
Buchholz: ψ_0(Ω_4^Ω_4^Ω_4)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,4)(6,4)(7,4)
Y sequence: (1,2,4,7,11,16,21,26,31)
#{&(&_2(&_3(&_4(1))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(0)))))
Buchholz: ψ_0(Ω_5)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)
Y sequence: (1,2,4,7,11,16,22)
#{&(&_2(&_3(&_4(&_4))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(Ω_4)))))
Buchholz: ψ_0(Ω_5·Ω_4)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,4)
Y sequence: (1,2,4,7,11,16,22,27)
#{&(&_2(&_3(&_4(&_5))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(Ω_5)))))
Buchholz: ψ_0(Ω_5^2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,5)
Y sequence: (1,2,4,7,11,16,22,28)
#{&(&_2(&_3(&_4(&_5^&_5))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(Ω_5^Ω_5)))))
Buchholz: ψ_0(Ω_5^Ω_5)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,5)(7,5)
Y sequence: (1,2,4,7,11,16,22,28,34)
#{&(&_2(&_3(&_4(&_5(1)))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(ψ_Ω_6(0))))))
Buchholz: ψ_0(Ω_6)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)
Y sequence: (1,2,4,7,11,16,22,29)
#{&(&_2(&_3(&_4(&_5(&_6)))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(ψ_Ω_6(Ω_6))))))
Buchholz: ψ_0(Ω_6^2)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,6)
Y sequence: (1,2,4,7,11,16,22,29,36)
#{&(&_2(&_3(&_4(&_5(&_6(1))))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(ψ_Ω_6(ψ_Ω_7(0)))))))
Buchholz: ψ_0(Ω_7)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)
Y sequence: (1,2,4,7,11,16,22,29,37)
#{&(&_2(&_3(&_4(&_5(&_6(&_7(1)))))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(ψ_Ω_6(ψ_Ω_7(ψ_Ω_8(0))))))))
Buchholz: ψ_0(Ω_8)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8)
Y sequence: (1,2,4,7,11,16,22,29,37,46)
#{&(&_2(&_3(&_4(&_5(&_6(&_7(&_8(1))))))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(ψ_Ω_6(ψ_Ω_7(ψ_Ω_8(ψ_Ω_9(0)))))))))
Buchholz: ψ_0(Ω_9)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8)(9,9)
Y sequence: (1,2,4,7,11,16,22,29,37,46,56)
#{&(&_2(&_3(&_4(&_5(&_6(&_7(&_8(&_9(1)))))))))}# has ordinal level:
Madore: ψ(ψ_Ω_2(ψ_Ω_3(ψ_Ω_4(ψ_Ω_5(ψ_Ω_6(ψ_Ω_7(ψ_Ω_8(ψ_Ω_9(ψ_Ω_10(0))))))))))
Buchholz: ψ_0(Ω_10)
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8)(9,9)(10,10)
Y sequence: (1,2,4,7,11,16,22,29,37,46,56,67)
And finally, the limit is: #{&(&_2(&_3(&_4(&_5(...&_(n-2)(&_(n-1)(&_n(1)))...))))}# = #{/}#, which has ordinal level:
Buchholz: ψ_0(Ω_ω) a.k.a. Buchholz's ordinal (BO)
BMS: (0,0,0)(1,1,1) a.k.a. the limit of pair sequence
Y sequence: (1,2,4,8)
Next, we define a handful of selected fundamental sequences of certain delimiters for all integer n from 0 to 3 (green for 1, blue for 2, and violet for 3) as follows:
#{&(1)}#[n]
0 → #
1 → #{&}#
2 → #{&^&}#
3 → #{&^&^&}#
(#{&(1)}#)^^#[n]
0 → (empty)
1 → #{&}#
2 → (#{&}#)^(#{&}#)
3 → #(#&}#)^(#{&}#)^(#{&}#)
(#{&(1)}#){&}#[n]
0 → (empty)
1 → #{&(1)}#
2 → (#{&(1)}#){#{&(1)}#}#
3 → (#{&(1)}#){(#{&(1)}#){#{&(1)#}#}#
(#{&(1)}#){&(1)}#[n]
0 → #{&(1)}#
1 → (#{&(1)}#){&}#
2 → (#{&(1)}#){&^&}#
3 → (#{&(1)}#){&^&^&}#
#{&(1)}#>#[n]
0 → (empty)
1 → #{&(1)}#
2 → (#{&(1)}#){&(1)}#
3 → ((#{&(1)}#){&(1)}#){&(1)}#
#{&(1)}##[n]
0 → (empty)
1 → #{&(1)}#
2 → #{&(1)}#>#{&(1)}#
3 → #{&(1)}#>#{&(1)}#>#{&(1)}#
#{&(1)+1}#[n]
0 → (empty)
1 → #{&(1)}#
2 → #{&(1)}#{&(1)}#
3 → #{&(1)}#{&(1)}#{&(1)}#
#{&(1)+&}#[n]
0 → (empty)
1 → #{&(1)}#
2 → #{&(1)+#{&(1)}#}#
3 → #{&(1)+#{&(1)+#{&(1)}#}#}#
#{&(1)+&^&}#[n]
0 → (empty)
1 → #{&(1)+&}#
2 → #{&(1)+&^#{&(1)+&}#}#
3 → #{&(1)+&^#{&(1)+&^#{&(1)+&}#}#}#
#{&(1)+&(1)}#[n]
0 → #{&(1)}#
1 → #{&(1)+&}#
2 → #{&(1)+&^&}#
3 → #{&(1)+&^&^&}#
#{&(1)*#}#[n]
0 → #
1 → #{&(1)}#
2 → #{&(1)+&(1)}#
3 → #{&(1)+&(1)+&(1)}#
#{&(1)*&}#[n]
0 → (empty)
1 → #{&(1)}#
2 → #{&(1)*#{&(1)}#}#
3 → #{&(1)*#{&(1)*#{&(1)}#}#}#
#{&(1)*&(1)}#[n]
0 → #{&(1)}#
1 → #{&(1)*&}#
2 → #{&(1)*&^&}#
3 → #{&(1)*&^&^&}#
#{&(1)^#}#[n]
0 → (empty)
1 → #{&(1)}#
2 → #{&(1)*&(1)}#
3 → #{&(1)*&(1)*&(1)}#
#{&(1)^&}#[n]
0 → (empty)
1 → #{&(1)}#
2 → #{&(1)^#{&(1)}#}#
3 → #{&(1)^#{&(1)^#{&(1)}#}#}#
#{&(1)^&(1)}#[n]
0 → #{&(1)}#
1 → #{&(1)^&}#
2 → #{&(1)^&^&}#
3 → #{&(1)^&^&^&}#
#{&(2)}#[n]
0 → #
1 → #{&(1)}#
2 → #{&(1)^&(1)}#
3 → #{&(1)^&(1)^&(1)}#
#{&(3)}#[n]
0 → #
1 → #{&(2)}#
2 → #{&(2)^&(2)}#
3 → #{&(2)^&(2)^&(2)}#
#{&(#)}#[n]
0 → #{&}#
1 → #{&(1)}#
2 → #{&(2)}#
3 → #{&(3)}#
#{&(&)}#[n]
0 → (empty)
1 → #{&(1)}#
2 → #{&(#{&(1)}#)}#
3 → #{&(#{&(#{&(1)}#)}#)}#
#{&(&&)}#[n]
0 → (empty)
1 → #{&(&)}#
2 → #{&(&#{&(&)}#)}#
3 → #{&(&#{&(&#{&(&)}#)}#)}#
#{&(&^&)}#[n]
0 → (empty)
1 → #{&(&)}#
2 → #{&(&^#{&(&)}#)}#
3 → #{&(&^#{&(&^#{&(&)}#)}#)}#
#{&(&(1))}#[n]
0 → #{&(1)}#
1 → #{&(&)}#
2 → #{&(&^&)}#
3 → #{&(&^&^&)}#
#{&(&(1))}#[n]
0 → #{&(1)}#
1 → #{&(&)}#
2 → #{&(&^&)}#
3 → #{&(&^&^&)}#
#{&(&_2)}#[n]
0 → #
1 → #{&(1)}#
2 → #{&(&(1))}#
3 → #{&(&(&(1)))}#
#{&(&_2+1)}#[n]
0 → #
1 → #{&(&_2)}#
2 → #{&(&_2)^&(&_2)}#
3 → #{&(&_2)^&(&_2)^&(&_2)}#
#{&(&_2+&)}#[n]
0 → (empty)
1 → #{&(&_2)}#
2 → #{&(&_2+#{&(&_2)}#)}#
3 → #{&(&_2+#{&(&_2+#{&(&_2)}#)}#)}#
#{&(&_2+&(1))}#[n]
0 → #{&(&_2)}#
1 → #{&(&_2+&)}#
2 → #{&(&_2+&^&)}#
3 → #{&(&_2+&^&^&)}#
#{&(&_2+&_2)}#[n]
0 → (empty)
1 → #{&(&_2)}#
2 → #{&(&_2+&(&_2))}#
3 → #{&(&_2+&(&_2+&(&_2)))}#
#{&(&_2*#)}#[n]
0 → #{&(1)}#
1 → #{&(&_2)}#
2 → #{&(&_2+&_2)}#
3 → #{&(&_2+&_2+&_2)}#
#{&(&_2*&)}#[n]
0 → (empty)
1 → #{&(&_2)}#
2 → #{&(&_2*#{&(&_2)}#)}#
3 → #{&(&_2*#{&(&_2*#{&(&_2)}#)}#)}#
#{&(&_2*&_2)}#[n]
0 → (empty)
1 → #{&(&_2)}#
2 → #{&(&_2*&(&_2))}#
3 → #{&(&_2*&(&_2*&(&_2)))}#
#{&(&_2^&_2)}#[n]
0 → (empty)
1 → #{&(&_2)}#
2 → #{&(&_2^&(&_2))}#
3 → #{&(&_2^&(&_2^&(&_2)))}#
#{&(&_2^&_2^&_2)}#[n]
0 → (empty)
1 → #{&(&_2^&_2)}#
2 → #{&(&_2^&_2^&(&_2^&_2))}#
3 → #{&(&_2^&_2^&(&_2^&_2^&(&_2^&_2)))}#
#{&(&_2(1))}#[n]
0 → #{&(1)}#
1 → #{&(&_2)}#
2 → #{&(&_2^&_2)}#
3 → #{&(&_2^&_2^&_2)}#
#{&(&_2(1)+1)}#[n]
0 → #
1 → #{&(&_2(1))}#
2 → #{&(&_2(1))^&(&_2(1))}#
3 → #{&(&_2(1))^&(&_2(1))^&(&_2(1))}#
#{&(&_2(1)+&)}#[n]
0 → (empty)
1 → #{&(&_2(1))}#
2 → #{&(&_2(1)+#{&(&_2(1))}#)}#
3 → #{&(&_2(1)+#{&(&_2(1)+#{&(&_2(1))}#)}#)}#
#{&(&_2(1)+&_2)}#[n]
0 → (empty)
1 → #{&(&_2(1))}#
2 → #{&(&_2(1)+&(&_2(1)))}#
3 → #{&(&_2(1)+&(&_2(1)+&(&_2(1))))}#
#{&(&_2(2))}#[n]
0 → #{&(1)}#
1 → #{&(&_2(1))}#
2 → #{&(&_2(1)^&_2(1))}#
3 → #{&(&_2(1)^&_2(1)^&_2(1))}#
#{&(&_2(&))}#[n]
0 → (empty)
1 → #{&(&_2(1))}#
2 → #{&(&_2(#{&(&_2(1))}#))}#
3 → #{&(&_2(#{&(&_2(#{&(&_2(1))}#))}#))}#
#{&(&_2(&_2))}#[n]
0 → #
1 → #{&(&_2(1))}#
2 → #{&(&_2(&(&_2(1))))}#
3 → #{&(&_2(&(&_2(&(&_2(1))))))}#
#{&(&_2(&_3))}#[n]
0 → #{&(1)}#
1 → #{&(&_2(1))}#
2 → #{&(&_2(&_2(1)))}#
3 → #{&(&_2(&_2(&_2(1))))}#
#{&(&_2(&_3^&_3))}#[n]
0 → #{&(1)}#
1 → #{&(&_2(&_3))}#
2 → #{&(&_2(&_3^&_2(&_3)))}#
3 → #{&(&_2(&_3^&_2(&_3^&_2(&_3))))}#
#{&(&_2(&_3(1)))}#[n]
0 → #{&(&_2(1))}#
1 → #{&(&_2(&_3))}#
2 → #{&(&_2(&_3^&_3))}#
3 → #{&(&_2(&_3^&_3^&_3))}#
#{&(&_2(&_3(&_4)))}#[n]
0 → #{&(&_2(1))}#
1 → #{&(&_2(&_3(1)))}#
2 → #{&(&_2(&_3(&_3(1))))}#
3 → #{&(&_2(&_3(&_3(&_3(1)))))}#
#{&(&_2(&_3(&_4(1))))}#[n]
0 → #{&(&_2(&_3(1)))}#
1 → #{&(&_2(&_3(&_4)))}#
2 → #{&(&_2(&_3(&_4^&_4)))}#
3 → #{&(&_2(&_3(&_4^&_4^&_4)))}#
#{&(&_2(&_3(&_4(&_5(1)))))}#[n]
0 → #{&(&_2(&_3(&_4(1))))}#
1 → #{&(&_2(&_3(&_4(&_5))))}#
2 → #{&(&_2(&_3(&_4(&_5^&_5))))}#
3 → #{&(&_2(&_3(&_4(&_5^&_5^&_5))))}#
#{/}#[n] (limit of x&E2)
0 → #
1 → #^^# (the one exception!)
2 → #{&(1)}#
3 → #{&(&_2(1))}#
4 → #{&(&_2(&_3(1)))}#
5 → #{&(&_2(&_3(&_4(1))))}#
Now it's the end of the analysis of the intended ordinal levels and fundamental sequences. And without further ado, let's go ahead and define the formal rules of the second system of the extended Collapsing-E notation (x&E2)!
From the rules of the basic Collapsing-E notation (&E):
Common rules
Rule 1. Base rule.
For k = 1 (with only one argument and no hyperions), we have E[x]n = x^n
Rule 2. Decomposition rule.
For L%(n-1) ≠ #^n (the last cascader is not in the form of #^n):
E[x]@a%b = E[x]@a%[b]a (@ indicates the unchanged remainder of the expression and %[b] is the fundamental sequence of %)
Rule 3. Termination rule.
For L%(n-1) = #^n (the last cascader is in the form of #^n), and the last argument is 1, it can be removed:
E[x]@a%1 = E[x]@a
Rule 4. Expansion rule.
For L%(n-1) = #^n and %k ≠ # (the last cascader is in the form of #^n but not the single hyperion):
E[x]@a%*#b = E[x]@a%a%*#(b-1)
Rule 5. Expansion rule.
Otherwise:
E[x]@a#b = E[x]@(E[x]@a#(b-1))
Decomposition rules
In addition the set of legal delimiters must be defined. Let & be the set of legal delimiters in xE^. The set is defined recursively:
We regard all elements of % in {n} as "transfinite n" (including countable and uncountable delimiters), for % ≥ #.
I. # is an element of %
II. If a,b are elements of % then a*b is an element of %
III. If a,b are elements of % then (a){n}(b) for "n ≥ 1 or transfinite n" is an element of %
IV. If a,b are elements of % and c is an element of %+, then (a){n}(b)>(c) for "n ≥ 1 or transfinite n" is an element of % for n>1.
V. If a is an element of % then a is an element of %+
VI. If a,b are elements of %+ then a+b is an element of %+
Lastly the decompositions of decomposable-delimiters must be defined. A delimiter, %, is decomposable (% is a member of %decomp), if and only if L(%) ≠ #^n.
Also, the decompositions of decomposable hyper-delimiters using "&" must be defined. A delimiter containing "&", %, is also decomposable (% is a member of %hdecomp, also known as an alternative of %decomp), if and only if L(%) ≠ #^n in {}.
The decompositions are defined as follows:
Case I. L = a^b, where a, b ∈ %:
A. When b = #:
I.A.1. %(a)^#[1] = %a
I.A.2. %(a)^#[n] = %a*(a)^#[n-1]
B. When b = k*#:
I.B.1. %(a)^(k*#)[1] = %(a)^(k)
I.B.2. %(a)^(k*#)[n] = %(a)^(k)*(a)^(k*#)[n-1]
C. When b ∈ %decomp:
%(a)^(b)[n] = %(a)^(b[n])
Case II. L = a{p}b, where a, b ∈ %, and (p > 1 or 0 < p < # in m+p, and m ≥ #):
(p is copies of multiple carets or a successor ordinal)
A. When b = #:
II.A.1. %(a){p}#[1] = %a
II.A.2. %(a){p}#[n] = %(a){p-1}((a){p}#[n-1])
B. When b = k*#:
II.B.1. %(a){p}(k*#)[1] = %a
II.B.2. %(a){p}(k*#)[n] = %(a){p-1}(k)>((a){p}(k*#)[n-1])
C. When b ∈ %decomp:
%(a){p}(b)[n] = %(a){p}(b[n])
Case III. L = a{p}b>c, where a, b ∈ %, c ∈ %+, and (p > 1 or 0 < p < # in m+p, and m ≥ #):
(p is copies of multiple carets or a successor ordinal)
A. When c = #:
III.A.1. %(a){p}(b)>#[1] = %(a){p}(b)
III.A.2. %(a){p}(b)>#[n] = %((a){p}(b)>#[n-1]){p}(b)
B. When c = k+#:
III.B.1. %(a){p}(b)>(k+#)[1] = %((a){p}(b)>(k)){p}(b)
III.B.2. %(a){p}(b)>#[n] = %((a){p}(b)>(k+#)[n-1]){p}(b)
C. When c ∈ %decomp:
%(a){p}(b)>(c)[n] = %(a){p}(b)>(c[n])
D. When c = k+d where k ∈ &+ and d ∈ %decomp:
%(a){p}(b)>(k+d)[n] = %(a){p}(b)>(k+d[n])
E. When c = d*# where d ∈ %:
III.E.1. %(a){p}(b)>(d*#)[1] = %(a){p}(b)>(d)
III.E.2. %(a){p}(b)>(d*#)[n] = %(a){p}(b)>(d+d*#)[n-1]
F. When c = k+d*# where k ∈ %+ and d ∈ %:
III.F.1. %(a){p}(b)>(k+d*#)[1] = %(a){p}(b)>(k+d)
III.F.2. %(a){p}(b)>(k+d*#)[n] = %(a){p}(b)>(k+d+d*#)[n-1]
Case IV. L = a{p}b, where L(p) ≠ m:
(L denotes the last sum of delimiters in {}, and m denotes the natural number)
A. When p = #:
&(a){#}#[n] = &(a)^^^^...^^^^# with n ^'s
B. When p = k+#:
&(a){k+#}#[n] = &(a){k+n}#
C. When p ∈ %decomp:
&(a){p}#[n] = &(a){p[n]}#
D. When p = k+c, k ∈ %+, c ∈ %decomp:
&(a){k+c}#[n] = &(a){k+c[n]}#
E. When p = c*# where c ∈ %+:
IV.E.1. &(a){c*#}#[1] = &(a){c}#
IV.E.2. &(a){c*#}#[n] = &(a){c+c*#}#[n-1]
F. When p = k+c*#, k ∈ %+, c ∈ %:
IV.E.1. &(a){k+c*#}#[1] = &(a){c}#
IV.E.2. &(a){k+c*#}#[n] = &(a){k+c+c*#}#[n-1]
Natural language equivalent of the formal rules:
The formal rules of the notation are very similar to those of Extended Cascading-E and Hyper-Extended Cascading-E notation extensions, except they are just the rewritten versions of the xE^ and #xE^ rules.
Case I. If L is an exponent operator (^), where a, b belong to previous member of %:
When the expression ends with copies of #: Copy the previous delimiter using hyper-product and one hyperion mark less after #. If the latter of the delimiter is not copies of #, or the last hyper-product is not copies of #, consult the rules for decomposing intermediate delimiter structures.
Case II. If L is in the form of a{p}b, where a, b ∈ %, and (p is natural number greater or equal to 2, or p is the successor ordinal):
Let the hyper operator iterate recursively via the up-arrow notation rules. If there are more than one hyperion after carets, or the last hyper-product is in the form of copies of hyperions, decompose the delimiter structure via the caret-tops, and with the identical delimiter structures. If the latter of the delimiter is not copies of #, consult the rules for decomposing intermediate delimiter structures.
Case III. If L in in the form of a{p}b>c, where a, b ∈ %, c ∈ %+, and (p is natural number greater or equal to 2, or p is the successor ordinal):
For each delimiter structures based on the hyper-operators from the least tetrational operator (^^#), with caret-tops, let it decompose into copies of hyperoperators from left to right. If there are plus signs followed by a single hyperion mark, let it decompose in the similar fashion, by removing the plus sign and a hyperion mark. If there are two or more consecutive hyperion marks, or the last hyper-product is in the form of copies of hyperions, use the hyperion-addition rule. If the latter of the delimiter is not copies of #, consult the rules for decomposing intermediate delimiter structures.
Case IV. If p in a{p}b is not a natural number or the successor ordinal
Consult the rule for decomposing delimiter structures inside curly braces, in a similar fashion to the rules on caret-tops.
From the rules introduced in the basic Collapsing-E notation:
Rule I. When c = &:
I1. %a{&}#[1] = %a
I2. %a{&}#[2] = %a{a}#
I3. %a{&}#[n] = %a{a{&}#[n-1]}# for n > 2
Rule II. When c = d+&
II1. %a{d+&}#[1] = %a{d}#
II2. %a{d+&}#[n] = %a{d+a{d+&}#[n-1]}#
Rule III. When c = d*& where d ∈ %+
III1. %a{d*&}#[1] = %a{d}#
III2. %a{d*&}#[n] = %a{d*a{d*&}#[n-1]}#
Rule IV. When c = k+d*&, k ∈ %+, d ∈ %:
IV1. %a{k+d*&}#[1] = %a{k+d}#
IV2. %a{k+d*&}#[n] = %a{k+d*a{k+d*&}#[n-1]}#
Rule V. When c = d^& where d ∈ %+:
V1. %a{d^&}#[1] = %a{d}#
V2. %a{d^&}#[n] = %a{d^a{d^&}#[n-1]}#
Rule VI. When c = k+d^&, k ∈ %+, d ∈ %:
VI1. %a{k+d^&}#[1] = %a{k+d}#
VI2. %a{k+d^&}#[n] = %a{k+d^a{k+d^&}#[n-1]}#
Rule VII.
When d^& where d ∈ %+, and the exponentiation rules of &, based off the Cascading-E notation rules, are applicable, consult the rules for case I in the decomposition rule:
VII1. %a{k*d}#[n] = %a{k*d[n]}#
VII2. %a{k^d}#[n] = %a{k^d[n]}#
VII3. %a{&^^#}#[n] = %a{&^^n}# = %a{&^&^&^...^&^&^&}# with n &'s
In a natural language equivalent:
I. For expressions with the delimiter structures containing just "&" delimiter inside curly braces {}, Repeat the left-hand side delimiter structure by nesting it inside curly braces recursively.
II. For expressions with the last hyper-operator-sum containing "&", let the left-hand side delimiter structure substitute recursively in place of &.
III. For ampersand-product and ampersand-exponent, consult the similar rules as the ampersand-sum decomposition rule.
And now, let's introduce a handful of formal rules of the extended Collapsing-E notation (system 2), a.k.a. x&E2. The valid form of the hyper-delimiters in extended Collapsing-E notation (system 2) is &(&_2(&_3(&_4(...&_(n-2)(&_(n-1)(&_n))...)))).
Let e be the active n-hyper-delimiter, and %A can be any delimiters that are influenced with the active &_k level (for instance, #{&(&_2(&_3+&_2))}# applies the entirety of the expression from &(e) for hyper-delimiter diagonalization, which in turn being &(&_2(&_3+&_2))):
Rule VIII. When e = &_k(f+g) where g is a natural number ≥ 1:
VIII1a. %A&_k(1)[1] = %A&_k
VIII1b. %A&_k(f+1)[1] = %A&_k(f)
VIII1c. %A&_k(f+g+1)[1] = %A&_k(f+g)
VIII2a. %A&_k(1)[n] = %A&_k^(&_k(1)[n-1]) for n > 1
VIII2b. %A&_k(f+1)[n] = %A&_k(f)^(&_k(f+1)[n-1]) for n > 1
VIII2c. %A&_k(f+g+1)[n] = %A&_k(f+g)^(&_k(f+g+1)[n-1]) for n > 1
*It can also be intuitively written as "&_k(f+1)[n] = &_k(f)^^n = &_k(f)^(&_k(f)^^(n-1))" as a part of the fundamental sequence denotion from the cascading-E notation, starting from tethrathoth.
Rule IX. In &_(k-1)(e) when e = &_k
IX1. %A&_(k-1)(&_k)[1] = %A&_(k-1)(1)
IX2. %A&_(k-1)(&_k)[n] = %A&_(k-1)(&_(k-1)(&_k)[n-1])
*This also applies for &_1 = &, where %A&_(k-1)(&)[1] is %A&_(k-1)(1), not %A&_(k-1)(0).
Rule X. In &_(k-1)(e) when e = f+&_k
X1. %A&_(k-1)(f+&_k)[1] = %A&_(k-1)(f)
X2. %A&_(k-1)(f+&_k)[n] = %A&_(k-1)(f+&_(k-1)(f+&_k)[n-1])
Rule XI. In &_(k-1)(e) when e = f*&_k, where f ∈ %+
XI1. %A&_(k-1)(f*&_k)[1] = %A&_(k-1)(f)
XI2. %A&_(k-1)(f*&_k)[n] = %A&_(k-1)(f*&_(k-1)(f*&_k)[n-1])
Rule XII. In &_(k-1)(e) when e = g+f*&_k, where g ∈ %+, f ∈ %
XII1. %A&_(k-1)(g+f*&_k)[1] = %A&_(k-1)(g+f)
XII2. %A&_(k-1)(g+f*&_k)[n] = %A&_(k-1)(g+f*&_(k-1)(g+f*&_k)[n-1])
Rule XIII. In &_(k-1)(e) when e = f^&_k, where f ∈ %+
XIII1. %A&_(k-1)(f^&_k)[1] = %A&_(k-1)(f)
XIII2. %A&_(k-1)(f^&_k)[n] = %A&_(k-1)(f^&_(k-1)(f^&_k)[n-1])
Rule XIV. In &_(k-1)(e) when e = g+f^&_k, where g ∈ %+, f ∈ %
XIV1. %A&_(k-1)(g+f^&_k)[1] = %A&_(k-1)(g+f)
XIV2. %A&_(k-1)(g+f^&_k)[n] = %A&_(k-1)(g+f^&_(k-1)(g+f^&_k)[n-1])
Rule XV.
When f^&_k where f ∈ %+, and the exponentiation rules of &_k, based off the Cascading-E notation rules, are applicable, consult the rules for case I in the decomposition rule:
XV1. %A&_k(g*f)[n] = %A&_k(g*f[n])
XV2. %A&_k(g^f)[n] = %A&_k(g^f[n])
In a natural language equivalent:
I. For successors in the form &_k(f+g), where g can be any natural numbers (not including zero or otherwise the g is degenerated), consult the rule on the power tower of &_k(f).
II. For expressions that use the &_k hyper-delimiter without the following parentheses that potentially support the &_(k+1) hyper-delimiter, either for the last hyper-operator sum (+), product (*), or power (^), let the left-hand side hyper-delimiter &_(k-1) structure substitute recursively in place of the active &_k, meaning that anything whatever the precedent delimiter &_k (in %A) is, the active &_k hyper-delimiter diagonalizes everything that belong to the &_(k-1) hyper-delimiter of the previous level.
III. More importantly, the &_n hyper-delimiter cannot appear inside of &_(n-2) hyper-delimiter or below. It can only appear in the form &(&_2(&_3(&_4(...)))) where &_n is the n-hyper-delimiter that increases one by one while nesting.
So, for instance, #{&(&_2(&_3)+&_2)}# is valid. On the other hand, #{&{&_2(&_4)}}# is invalid as &_4 requires to be nested inside of &_3(n), which violates the rule III of the aforementioned natural language equivalent.
And that's the formal definition of the second system of the extended Collapsing-E notation. It is not difficult to define, since the additional rules are effectively derived from the existing rules of the basic Collapsing-E notation, except it allows the subscripts next to & and uses &_n in the rules above.
And now, we have the system that's grows all the way to Buchholz's ordinal, which also corresponds to (0,0,0)(1,1,1) in Bashicu matrix system (BMS, also the limit of pair sequence system) or (1,2,4,8) in Y sequence.
E4#{&(1)+&^&&}#4
= E4#{&(1)+&^&&}#[4]4
= E4#{&(1)+&^&#{&(1)+&^&&}#[3]}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&&}#[2]}#}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&&}#[1]}#}#}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&}#}#}#}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&}#[4]}#}#}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&#{&(1)+&^#{&(1)+&^&}#[3]}#}#}#}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&#{&(1)+&^#{&(1)+&^#{&(1)+&^&}#[2]}#}#}#}#}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&#{&(1)+&^#{&(1)+&^#{&(1)+&^#{&(1)+&^&}#[1]}#}#}#}#}#}#4
= E4#{&(1)+&^&#{&(1)+&^&#{&(1)+&^&#{&(1)+&^#{&(1)+&^#{&(1)+&^#{&(1)+&}#}#}#}#}#}#}#4
= ...
E4#{&(&+1)}#4
= E4#{&(&+1)}#[4]4
= E4#{&(&+1)[4]}#4
= E4#{&(&)^&(&+1)[3]}#4
= E4#{&(&)^&(&)^&(&+1)[2]}#4
= E4#{&(&)^&(&)^&(&)^&(&+1)[1]}#4
= E4#{&(&)^&(&)^&(&)^&(&)}#4
= E4#{&(&)^&(&)^&(&)^&(&)}#[4]4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(&)}#[3])}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(&)}#[2])}#)}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(&)}#[1])})}#)}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(1)}#)})}#)}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(1)[4]}#)})}#)}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&^&(1)[3]}#)})}#)}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&^&^&(1)[2]}#)})}#)}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&^&^&^&(1)[1]}#)})}#)}#4
= E4#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&(#{&(&)^&(&)^&(&)^&^&^&^&}#)})}#)}#4
= ...
E4#{&(&_2^&+&_2)}#4
= E4#{&(&_2^&+&_2)}#[4]4
= E4#{&(&_2^&+&_2)[4]}#4
= E4#{&(&_2^&+&(&_2^&+&_2)[3])}#4
= E4#{&(&_2^&+&(&_2^&+&(&_2^&+&_2)[2]))}#4
= E4#{&(&_2^&+&(&_2^&+&(&_2^&+&(&_2^&+&_2)[1])))}#4
= E4#{&(&_2^&+&(&_2^&+&(&_2^&+&(&_2^&+&))))}#4
= ...
E4#{&(&_2(&_3(&_4^&_4^&_3)))}#4
= E4#{&(&_2(&_3(&_4^&_4^&_3)))}#[4]4
= E4#{&(&_2(&_3(&_4^&_4^&_3)))[4]}#4
= E4#{&(&_2(&_3(&_4^&_4^&_3))[4])}#4
= E4#{&(&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_3))[3])))}#4
= E4#{&(&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_3))[2])))))}#4
= E4#{&(&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_3))[1])))))))}#4
= E4#{&(&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4^&_2(&_3(&_4^&_4))))))))}#4
= ...