The previous definition attempt of the extended collapsing-E notation was a kind of awkward. I need to re-designate the fundamental sequences of the particular delimiters.
Here, I am going to designate the new fundamental sequences with the entirely new rules.
[1] #{&_2}#
N = 1: #{&}#
N = 2: #{&^&}#
N = 3: #{&^&^&}#
FGH ordinal level: ψ0(Ω_2)
[2] (#{&_2}#)^^#
N = 1: #{&_2}#
N = 2: (#{&_2}#)^(#{&_2}#)
N = 3: (#{&_2}#)^(#{&_2}#)^(#{&_2}#)
FGH ordinal level: ε(ψ0(Ω_2) + 1)
[3] (#{&_2}#)^^##
N = 1: (#{&_2}#)^^#
N = 2: (#{&_2}#)^^#>(#{&_2}#)^^#
N = 3: (#{&_2}#)^^#>(#{&_2}#)^^#>(#{&_2}#)^^#
FGH ordinal level: ζ(ψ0(Ω_2) + 1)
[4] (#{&_2}#)^^^#
N = 1: (#{&_2}#)
N = 2: (#{&_2}#)^^(#{&_2}#)
N = 3: (#{&_2}#)^^(#{&_2}#)^^(#{&_2}#)
FGH ordinal level: Γ(ψ0(Ω_2) + 1)
[5] (#{&_2}#){#}#
N = 1: (#{&_2}#)^#
N = 2: (#{&_2}#)^^#
N = 3: (#{&_2}#)^^^#
FGH ordinal level: φ(ω, 0, ψ0(Ω_2) + 1)
[6] (#{&_2}#){&}#
N = 1: #{&_2}#
N = 2: (#{&_2}#){#{&_2}#}#
N = 3: (#{&_2}#){(#{&_2}#){#{&_2}#}#}#
FGH ordinal level: φ(1, 0, 0, ψ0(Ω_2) + 1)
[7] (#{&_2}#){&&}#
N = 1: (#{&_2}#){&}#
N = 2: (#{&_2}#){&(#{&_2}#){&}#}#
N = 3: (#{&_2}#){&(#{&_2}#){&(#{&_2}#){&}#}#}#
FGH ordinal level: φ(1, 0, 0, 0, ψ0(Ω_2) + 1)
[8] (#{&_2}#){&^#}#
N = 1: (#{&_2}#){&}#
N = 2: (#{&_2}#){&&}#
N = 3: (#{&_2}#){&&&}#
FGH ordinal level: φ(1 @{ω} ψ0(Ω_2) + 1)
[9] (#{&_2}#){&^&}#
N = 1: (#{&_2}#){&}#
N = 2: (#{&_2}#){&^(#{&_2}#){&}#}#
N = 3: (#{&_2}#){&^(#{&_2}#){&^(#{&_2}#){&}#}#}#
FGH ordinal level: φ(1 @{1, 0} ψ0(Ω_2) + 1)
[10] (#{&_2}#){(&_2)_1}#
N = 1: (#{&_2}#){&}#
N = 2: (#{&_2}#){&^&}#
N = 3: (#{&_2}#){&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1))
[11] ((#{&_2}#){(&_2)_1}#)^^#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: ((#{&_2}#){(&_2)_1}#)^((#{&_2}#){(&_2)_1}#)
N = 3: ((#{&_2}#){(&_2)_1}#)^((#{&_2}#){(&_2)_1}#)^((#{&_2}#){(&_2)_1}#)
FGH ordinal level: ε(ψ0(Ω_2 + ε(Ω + 1)) + 1)
[12] ((#{&_2}#){(&_2)_1}#){(&_2)_1}#
N = 1: ((#{&_2}#){(&_2)_1}#){&}#
N = 2: ((#{&_2}#){(&_2)_1}#){&^&}#
N = 3: ((#{&_2}#){(&_2)_1}#){&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·2)
[13] (#{&_2}#){(&_2)_1}#>#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: ((#{&_2}#){(&_2)_1}#){(&_2)_1}#
N = 3: (((#{&_2}#){(&_2)_1}#){(&_2)_1}#){(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ω)
[14] ((#{&_2}#){(&_2)_1}#>#){(&_2)_1}#
N = 1: ((#{&_2}#){(&_2)_1}#>#){&}#
N = 2: ((#{&_2}#){(&_2)_1}#>#){&^&}#
N = 3: ((#{&_2}#){(&_2)_1}#>#){&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·(ω + 1))
[15] (#{&_2}#){(&_2)_1}#>(#+#)
N = 1: ((#{&_2}#){(&_2)_1}#>#){(&_2)_1}#
N = 2: (((#{&_2}#){(&_2)_1}#>#){(&_2)_1}#){(&_2)_1}#
N = 3: ((((#{&_2}#){(&_2)_1}#>#){(&_2)_1}#){(&_2)_1}#){(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ω·2)
[16] (#{&_2}#){(&_2)_1}#>##
N = 1: (#{&_2}#){(&_2)_1}#>#
N = 2: (#{&_2}#){(&_2)_1}#>(#+#)
N = 3: (#{&_2}#){(&_2)_1}#>(#+#+#)
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ω^2)
[17] (#{&_2}#){(&_2)_1}#>#^#
N = 1: (#{&_2}#){(&_2)_1}#>#
N = 2: (#{&_2}#){(&_2)_1}#>##
N = 3: (#{&_2}#){(&_2)_1}#>###
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ω^ω)
[18] (#{&_2}#){(&_2)_1}#>#^^#
N = 1: (#{&_2}#){(&_2)_1}#>#
N = 2: (#{&_2}#){(&_2)_1}#>#^#
N = 3: (#{&_2}#){(&_2)_1}#>#^#^#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ε0)
[19] (#{&_2}#){(&_2)_1}#>#{&_2}#
N = 1: (#{&_2}#){(&_2)_1}#>#{&}#
N = 2: (#{&_2}#){(&_2)_1}#>#{&^&}#
N = 3: (#{&_2}#){(&_2)_1}#>#{&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ψ0(Ω_2))
[20] (#{&_2}#){(&_2)_1}#>(#{&_2}#)^^#
N = 1: (#{&_2}#){(&_2)_1}#>#{&_2}#
N = 2: (#{&_2}#){(&_2)_1}#>(#{&_2}#)^(#{&_2}#)
N = 3: (#{&_2}#){(&_2)_1}#>(#{&_2}#)^(#{&_2}#)^(#{&_2}#)
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ε(ψ0(Ω_2) + 1))
[21] (#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}#
N = 1: (#{&_2}#){(&_2)_1}#>(#{&_2}#){&}#
N = 2: (#{&_2}#){(&_2)_1}#>(#{&_2}#){&^&}#
N = 3: (#{&_2}#){(&_2)_1}#>(#{&_2}#){&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·ψ0(Ω_2 + ε(Ω + 1)))
[22] (#{&_2}#){(&_2)_1}##
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}#
N = 3: (#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω)
[23] ((#{&_2}#){(&_2)_1}##){(&_2)_1}#
N = 1: ((#{&_2}#){(&_2)_1}#){&}#
N = 2: ((#{&_2}#){(&_2)_1}#){&^&}#
N = 3: ((#{&_2}#){(&_2)_1}#){&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·(Ω + 1))
[24] ((#{&_2}#){(&_2)_1}##){(&_2)_1}##
N = 1: ((#{&_2}#){(&_2)_1}##){(&_2)_1}#
N = 2: ((#{&_2}#){(&_2)_1}##){(&_2)_1}#>((#{&_2}#){(&_2)_1}##){(&_2)_1}#
N = 3: ((#{&_2}#){(&_2)_1}##){(&_2)_1}#>((#{&_2}#){(&_2)_1}##){(&_2)_1}#>((#{&_2}#){(&_2)_1}##){(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω·2)
[25] (#{&_2}#){(&_2)_1}##>#
N = 1: (#{&_2}#){(&_2)_1}##
N = 2: ((#{&_2}#){(&_2)_1}##){(&_2)_1}##
N = 3: (((#{&_2}#){(&_2)_1}##){(&_2)_1}##){(&_2)_1}##
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω·ω)
[26] (#{&_2}#){(&_2)_1}##>#{&_2}#
N = 1: (#{&_2}#){(&_2)_1}##>#{&}#
N = 2: (#{&_2}#){(&_2)_1}##>#{&^&}#
N = 3: (#{&_2}#){(&_2)_1}##>#{&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω·ψ0(Ω_2))
[27] (#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}#
N = 1: (#{&_2}#){(&_2)_1}##>(#{&_2}#){&}#
N = 2: (#{&_2}#){(&_2)_1}##>(#{&_2}#){&^&}#
N = 3: (#{&_2}#){(&_2)_1}##>(#{&_2}#){&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω·ψ0(Ω_2 + ε(Ω + 1)))
[28] (#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}##
N = 1: (#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}#
N = 3: (#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω·ψ0(Ω_2 + ε(Ω + 1)·Ω))
[29] (#{&_2}#){(&_2)_1}###
N = 1: (#{&_2}#){(&_2)_1}##
N = 2: (#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}##
N = 3: (#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}##>(#{&_2}#){(&_2)_1}##
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^2)
[30] (#{&_2}#){(&_2)_1}#^#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1}##
N = 3: (#{&_2}#){(&_2)_1}###
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^ω)
[31] (#{&_2}#){(&_2)_1}#^^#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1}#^#
N = 3: (#{&_2}#){(&_2)_1}#^#^#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^ε0)
[32] (#{&_2}#){(&_2)_1}#{(&_2)_1}#
N = 1: (#{&_2}#){(&_2)_1}#{&}#
N = 2: (#{&_2}#){(&_2)_1}#{&^&}#
N = 3: (#{&_2}#){(&_2)_1}#{&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^ψ0(Ω_2))
[33] (#{&_2}#){(&_2)_1 + 1}#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1}#{(&_2)_1}#
N = 3: (#{&_2}#){(&_2)_1}#{(&_2)_1}#{(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω)
[34] (#{&_2}#){(&_2)_1 + 1}##
N = 1: (#{&_2}#){(&_2)_1 + 1}#
N = 2: (#{&_2}#){(&_2)_1 + 1}#>(#{&_2}#){(&_2)_1 + 1}#
N = 3: (#{&_2}#){(&_2)_1 + 1}#>(#{&_2}#){(&_2)_1 + 1}#>(#{&_2}#){(&_2)_1 + 1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω + 1))
[35] (#{&_2}#){(&_2)_1 + 1}#^#
N = 1: (#{&_2}#){(&_2)_1 + 1}#
N = 2: (#{&_2}#){(&_2)_1 + 1}##
N = 3: (#{&_2}#){(&_2)_1 + 1}###
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω + ω))
[36] (#{&_2}#){(&_2)_1 + 1}#{(&_2)_1}#
N = 1: (#{&_2}#){(&_2)_1 + 1}#{&}#
N = 2: (#{&_2}#){(&_2)_1 + 1}#{&^&}#
N = 3: (#{&_2}#){(&_2)_1 + 1}#{&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω + ψ0(Ω_2)))
[37] (#{&_2}#){(&_2)_1 + 1}#{(&_2)_1 + 1}#
N = 1: (#{&_2}#){(&_2)_1 + 1}#{(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1 + 1}#{(&_2)_1}#{(&_2)_1}#
N = 3: (#{&_2}#){(&_2)_1 + 1}#{(&_2)_1}#{(&_2)_1}#{(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω + ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω)))
[38] (#{&_2}#){(&_2)_1 + 2}#
N = 1: (#{&_2}#){(&_2)_1 + 1}#
N = 2: (#{&_2}#){(&_2)_1 + 1}#{(&_2)_1 + 1}#
N = 3: (#{&_2}#){(&_2)_1 + 1}#{(&_2)_1 + 1}#{(&_2)_1 + 1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω·2))
[39] (#{&_2}#){(&_2)_1 + #}#
N = 1: (#{&_2}#){(&_2)_1 + 1}#
N = 2: (#{&_2}#){(&_2)_1 + 2}#
N = 3: (#{&_2}#){(&_2)_1 + 3}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω·ω))
[40] (#{&_2}#){(&_2)_1 + #{&_2}#}#
N = 1: (#{&_2}#){(&_2)_1 + #{&}#}#
N = 2: (#{&_2}#){(&_2)_1 + #{&^&}#}#
N = 3: (#{&_2}#){(&_2)_1 + #{&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω·ψ0(Ω_2)))
[41] (#{&_2}#){(&_2)_1 + (#{&_2}#){(&_2)_1}#}#
N = 1: (#{&_2}#){(&_2)_1 + (#{&_2}#){&}#}#
N = 2: (#{&_2}#){(&_2)_1 + (#{&_2}#){&^&}#}#
N = 3: (#{&_2}#){(&_2)_1 + (#{&_2}#){&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^(Ω·ψ0(Ω_2 + ε(Ω + 1))))
[42] (#{&_2}#){(&_2)_1 + &}#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1 + (#{&_2}#){(&_2)_1}#}#
N = 3: (#{&_2}#){(&_2)_1 + (#{&_2}#){(&_2)_1 + (#{&_2}#){(&_2)_1}#}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^2)
[43] (#{&_2}#){(&_2)_1 + &^#}#
N = 1: (#{&_2}#){(&_2)_1 + &}#
N = 2: (#{&_2}#){(&_2)_1 + &&}#
N = 3: (#{&_2}#){(&_2)_1 + &&&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^ω)
[44] (#{&_2}#){(&_2)_1 + &^#{&_2}#}#
N = 1: (#{&_2}#){(&_2)_1 + &^#{&}#}#
N = 2: (#{&_2}#){(&_2)_1 + &^#{&^&}#}#
N = 3: (#{&_2}#){(&_2)_1 + &^#{&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^ψ0(Ω_2))
[45] (#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1}#}#
N = 1: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){&}#}#
N = 2: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){&^&}#}#
N = 3: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^ψ0(Ω_2 + ε(Ω + 1)))
[46] (#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1 + &}#}#
N = 1: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1}#}#
N = 2: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1 + (#{&_2}#){(&_2)_1}#}#}#
N = 3: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1 + (#{&_2}#){(&_2)_1 + (#{&_2}#){(&_2)_1}#}#}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^2))
[47] (#{&_2}#){(&_2)_1 + &^&}#
N = 1: (#{&_2}#){(&_2)_1 + &}#
N = 2: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1 + &}#}#
N = 3: (#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1 + &^(#{&_2}#){(&_2)_1 + &}#}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^Ω)
[48] (#{&_2}#){(&_2)_1 + &^&^&}#
N = 1: (#{&_2}#){(&_2)_1 + &^&}#
N = 2: (#{&_2}#){(&_2)_1 + &^&^(#{&_2}#){(&_2)_1 + &^&}#}#
N = 3: (#{&_2}#){(&_2)_1 + &^&^(#{&_2}#){(&_2)_1 + &^&^(#{&_2}#){(&_2)_1 + &^&}#}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)·Ω^Ω^Ω^Ω)
[49] (#{&_2}#){(&_2)_1 + (&_2)_1}#
N = 1: (#{&_2}#){(&_2)_1 + &}#
N = 2: (#{&_2}#){(&_2)_1 + &^&}#
N = 3: (#{&_2}#){(&_2)_1 + &^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^2)
[50] (#{&_2}#){(&_2)_1*#}#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1 + (&_2)_1}#
N = 3: (#{&_2}#){(&_2)_1 + (&_2)_1 + (&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ω)
[51] (#{&_2}#){(&_2)_1*#{&_2}#}#
N = 1: (#{&_2}#){(&_2)_1*#{&}#}#
N = 2: (#{&_2}#){(&_2)_1*#{&^&}#}#
N = 3: (#{&_2}#){(&_2)_1*#{&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ψ0(Ω_2))
[52] (#{&_2}#){(&_2)_1*(#{&_2}#){(&_2)_1}#}#
N = 1: (#{&_2}#){(&_2)_1*(#{&_2}#){&}#}#
N = 2: (#{&_2}#){(&_2)_1*(#{&_2}#){&^&}#}#
N = 3: (#{&_2}#){(&_2)_1*(#{&_2}#){&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ψ0(Ω_2 + ε(Ω + 1)))
[53] (#{&_2}#){(&_2)_1*&}#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1*(#{&_2}#){(&_2)_1}#}#
N = 3: (#{&_2}#){(&_2)_1*(#{&_2}#){(&_2)_1*(#{&_2}#){(&_2)_1}#}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^Ω)
[54] (#{&_2}#){(&_2)_1*(&_2)_1}#
N = 1: (#{&_2}#){(&_2)_1*&}#
N = 2: (#{&_2}#){(&_2)_1*&^&}#
N = 3: (#{&_2}#){(&_2)_1*&^&^&}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1))
[55] (#{&_2}#){(&_2)_1^#}#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1*(&_2)_1}#
N = 3: (#{&_2}#){(&_2)_1*(&_2)_1*(&_2)_1}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ω)
[56] (#{&_2}#){(&_2)_1^#{&_2}#}#
N = 1: (#{&_2}#){(&_2)_1^#{&}#}#
N = 2: (#{&_2}#){(&_2)_1^#{&^&}#}#
N = 3: (#{&_2}#){(&_2)_1^#{&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ψ0(Ω_2))
[57] (#{&_2}#){(&_2)_1^(#{&_2}#){(&_2)_1}#}#
N = 1: (#{&_2}#){(&_2)_1^(#{&_2}#){&}#}#
N = 2: (#{&_2}#){(&_2)_1^(#{&_2}#){&^&}#}#
N = 3: (#{&_2}#){(&_2)_1^(#{&_2}#){&^&^&}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ψ0(Ω_2 + ε(Ω + 1))
[58] (#{&_2}#){(&_2)_1^(&_2)_1}#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1^(#{&_2}#){(&_2)_1}#}#
N = 3: (#{&_2}#){(&_2)_1^(#{&_2}#){(&_2)_1^(#{&_2}#){(&_2)_1}#}#}#
FGH ordinal level: ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1))
Now we inch into the new level. It gets even crazier...
[59] (#{&_2}#){&_2}#
N = 1: (#{&_2}#){(&_2)_1}#
N = 2: (#{&_2}#){(&_2)_1^(&_2)_1}#
N = 3: (#{&_2}#){(&_2)_1^(&_2)_1^(&_2)_1}#
FGH ordinal level: ψ0(Ω_2·2)
[60] ((#{&_2}#){&_2}#){(&_2)_1}#
N = 1: ((#{&_2}#){&_2}#){&}#
N = 2: ((#{&_2}#){&_2}#){&^&}#
N = 3: ((#{&_2}#){&_2}#){&^&^&}#
FGH ordinal level: ψ0(Ω_2·2 + ε(Ω + 1))
[61] ((#{&_2}#){&_2}#){(&_2)_1<[#>2]}#
N = 1: ((#{&_2}#){&_2}#){(&_2)_1}#
N = 2: ((#{&_2}#){&_2}#){(&_2)_1^(&_2)_1}#
N = 3: ((#{&_2}#){&_2}#){(&_2)_1^(&_2)_1^(&_2)_1}#
FGH ordinal level: ψ0(Ω_2·2 + ε(Ω + 2))
[62] ((#{&_2}#){&_2}#){(&_2)_1<[#>2] + (&_2)_1<[#>2]}#
N = 1: ((#{&_2}#){&_2}#){(&_2)_1<[#>2] + (&_2)_1}#
N = 2: ((#{&_2}#){&_2}#){(&_2)_1<[#>2] + (&_2)_1^(&_2)_1}#
N = 3: ((#{&_2}#){&_2}#){(&_2)_1<[#>2] + (&_2)_1^(&_2)_1^(&_2)_1}#
FGH ordinal level: ψ0(Ω_2·2 + ε(Ω + 2)^2)
[63] ((#{&_2}#){&_2}#){&_2}#
N = 1: ((#{&_2}#){&_2}#){(&_2)_1<[#>2]}#
N = 2: ((#{&_2}#){&_2}#){(&_2)_1<[#>2]^(&_2)_1<[#>2]}#
N = 3: ((#{&_2}#){&_2}#){(&_2)_1<[#>2]^(&_2)_1<[#>2]^(&_2)_1<[#>2]}#
FGH ordinal level: ψ0(Ω_2·3)
[64] (((#{&_2}#){&_2}#){&_2}#){&_2}#
N = 1: (((#{&_2}#){&_2}#){&_2}#){(&_2)_1<[#>3]}#
N = 2: (((#{&_2}#){&_2}#){&_2}#){(&_2)_1<[#>3]^(&_2)_1<[#>3]}#
N = 3: (((#{&_2}#){&_2}#){&_2}#){(&_2)_1<[#>3]^(&_2)_1<[#>3]^(&_2)_1<[#>3]}#
FGH ordinal level: ψ0(Ω_2·4)
[65] #{&_2}#>#
N = 1: #{&_2}#
N = 2: (#{&_2}#){&_2}#
N = 3: ((#{&_2}#){&_2}#){&_2}#
FGH ordinal level: ψ0(Ω_2·ω)
[66] (#{&_2}#>#){(&_2)_1}#
N = 1: (#{&_2}#>#){&}#
N = 2: (#{&_2}#>#){&^&}#
N = 3: (#{&_2}#>#){&^&^&}#
FGH ordinal level: ψ0(Ω_2·ω + ε(Ω + 1))
[67] (#{&_2}#>#){(&_2)_1<[#>2]}#
N = 1: (#{&_2}#>#){(&_2)_1}#
N = 2: (#{&_2}#>#){(&_2)_1^(&_2)_1}#
N = 3: (#{&_2}#>#){(&_2)_1^(&_2)_1^(&_2)_1}#
FGH ordinal level: ψ0(Ω_2·ω + ε(Ω + 2))
[68] (#{&_2}#>#){(&_2)_1<[#>#]}#
N = 1: (#{&_2}#>#){(&_2)_1}#
N = 2: (#{&_2}#>#){(&_2)_1<[#>2]}#
N = 3: (#{&_2}#>#){(&_2)_1<[#>3]}#
FGH ordinal level: ψ0(Ω_2·ω + ε(Ω + ω))
[69] (#{&_2}#>#){&_2}#
N = 1: (#{&_2}#>#){(&_2)_1<[#>#]}#
N = 2: (#{&_2}#>#){(&_2)_1<[#>#]^(&_2)_1<[#>#]}#
N = 3: (#{&_2}#>#){(&_2)_1<[#>#]^(&_2)_1<[#>#]^(&_2)_1<[#>#]}#
FGH ordinal level: ψ0(Ω_2·(ω + 1))
[70] #{&_2}#>(#+#)
N = 1: (#{&_2}#>#){&_2}#
N = 2: ((#{&_2}#>#){&_2}#){&_2}#
N = 3: (((#{&_2}#>#){&_2}#){&_2}#){&_2}#
FGH ordinal level: ψ0(Ω_2·ω·2)
[71] #{&_2}#>##
N = 1: #{&_2}#>#
N = 2: #{&_2}#>(#+#)
N = 3: #{&_2}#>(#+#+#)
FGH ordinal level: ψ0(Ω_2·ω^2)
[72] #{&_2}#>#^#
N = 1: #{&_2}#>#
N = 2: #{&_2}#>##
N = 3: #{&_2}#>###
FGH ordinal level: ψ0(Ω_2·ω^ω)
[73] #{&_2}#>#^^#
N = 1: #{&_2}#>#
N = 2: #{&_2}#>#^#
N = 3: #{&_2}#>#^#^#
FGH ordinal level: ψ0(Ω_2·ε0)
[74] #{&_2}#>#{&_2}#
N = 1: #{&_2}#>#{&}#
N = 2: #{&_2}#>#{&^&}#
N = 3: #{&_2}#>#{&^&^&}#
FGH ordinal level: ψ0(Ω_2·ψ0(Ω_2))
Now we have two hyperions next to the bracket {}...
[75] #{&_2}##
N = 1: #{&_2}#
N = 2: #{&_2}#>#{&_2}#
N = 3: #{&_2}#>#{&_2}#>#{&_2}#
FGH ordinal level: ψ0(Ω_2·Ω)
[76] (#{&_2}##)^^#
N = 1: #{&_2}##
N = 2: (#{&_2}##)^(#{&_2}##)
N = 3: (#{&_2}##)^(#{&_2}##)^(#{&_2}##)
FGH ordinal level: ε(ψ0(Ω_2·Ω) + 1)
[77] (#{&_2}##){(&_2)_1}#
N = 1: (#{&_2}##){&}#
N = 2: (#{&_2}##){&^&}#
N = 3: (#{&_2}##){&^&^&}#
FGH ordinal level: ψ0(Ω_2·Ω + ε(Ω + 1))
[78] (#{&_2}##){(&_2)_1<[##]}#
N = 1: (#{&_2}##){(&_2)_1}#
N = 2: (#{&_2}##){(&_2)_1<[#>(#{&_2}##){(&_2)_1}#]}#
N = 3: (#{&_2}##){(&_2)_1<[#>(#{&_2}##){(&_2)_1<[#>(#{&_2}##){(&_2)_1}#]}#]}#
FGH ordinal level: ψ0(Ω_2·Ω + ε(Ω·2))
[79] (#{&_2}##){&_2}#
N = 1: (#{&_2}##){(&_2)_1<[##]}#
N = 2: (#{&_2}##){(&_2)_1<[##]^(&_2)_1<[##]}#
N = 3: (#{&_2}##){(&_2)_1<[##]^(&_2)_1<[##]^(&_2)_1<[##]}#
FGH ordinal level: ψ0(Ω_2·(Ω + 1))
[80] (#{&_2}##){&_2}##
N = 1: (#{&_2}##){&_2}#
N = 2: (#{&_2}##){&_2}#>(#{&_2}##){&_2}#
N = 3: (#{&_2}##){&_2}#>(#{&_2}##){&_2}#>(#{&_2}##){&_2}#
FGH ordinal level: ψ0(Ω_2·Ω·2)
[81] #{&_2}##>#
N = 1: #{&_2}##
N = 2: (#{&_2}##){&_2}##
N = 3: ((#{&_2}##){&_2}##){&_2}##
FGH ordinal level: ψ0(Ω_2·Ω·ω)
[82] #{&_2}##>#{&_2}#
N = 1: #{&_2}##>#{&}#
N = 2: #{&_2}##>#{&^&}#
N = 3: #{&_2}##>#{&^&^&}#
FGH ordinal level: ψ0(Ω_2·Ω·ψ0(Ω_2))
[83] #{&_2}##>#{&_2}##
N = 1: #{&_2}##>#{&_2}#
N = 2: #{&_2}##>#{&_2}#>#{&_2}#
N = 3: #{&_2}##>#{&_2}#>#{&_2}#>#{&_2}#
FGH ordinal level: ψ0(Ω_2·Ω·ψ0(Ω_2·Ω))
[84] #{&_2}###
N = 1: #{&_2}##
N = 2: #{&_2}##>#{&_2}##
N = 3: #{&_2}##>#{&_2}##>#{&_2}##
FGH ordinal level: ψ0(Ω_2·Ω^2)
[85] #{&_2}#^#
N = 1: #{&_2}#
N = 2: #{&_2}##
N = 3: #{&_2}###
FGH ordinal level: ψ0(Ω_2·Ω^ω)
[86] #{&_2}#{&_2}#
N = 1: #{&_2}#{&}#
N = 2: #{&_2}#{&^&}#
N = 3: #{&_2}#{&^&^&}#
FGH ordinal level: ψ0(Ω_2·Ω^ψ0(Ω_2))
[87] #{&_2 + 1}#
N = 1: #{&_2}#
N = 2: #{&_2}#{&_2}#
N = 3: #{&_2}#{&_2}#{&_2}#
FGH ordinal level: ψ0(Ω_2·Ω^Ω)
[88] #{&_2 + 2}#
N = 1: #{&_2 + 1}#
N = 2: #{&_2 + 1}#{&_2 + 1}#
N = 3: #{&_2 + 1}#{&_2 + 1}#{&_2 + 1}#
FGH ordinal level: ψ0(Ω_2·Ω^(Ω·2))
[89] #{&_2 + #}#
N = 1: #{&_2 + 1}#
N = 2: #{&_2 + 2}#
N = 3: #{&_2 + 3}#
FGH ordinal level: ψ0(Ω_2·Ω^(Ω·ω))
[90] #{&_2 + &}#
N = 1: #{&_2}#
N = 2: #{&_2 + #{&_2}#}#
N = 3: #{&_2 + #{&_2 + #{&_2}#}#}#
FGH ordinal level: ψ0(Ω_2·Ω^Ω^2)
[91] #{&_2 + &^#}#
N = 1: #{&_2 + &}#
N = 2: #{&_2 + &&}#
N = 3: #{&_2 + &&&}#
FGH ordinal level: ψ0(Ω_2·Ω^Ω^ω)
[92] #{&_2 + &^&}#
N = 1: #{&_2 + &}#
N = 2: #{&_2 + &^#{&_2 + &}#}#
N = 3: #{&_2 + &^#{&_2 + &^#{&_2 + &}#}#}#
FGH ordinal level: ψ0(Ω_2·Ω^Ω^Ω)
[93] #{&_2 + (&_2)_1}#
N = 1: #{&_2 + &}#
N = 2: #{&_2 + &^&}#
N = 3: #{&_2 + &^&^&}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1))
[94] (#{&_2 + (&_2)_1}#){&_2 + (&_2)_1}#
N = 1: (#{&_2 + (&_2)_1}#){&_2 + &}#
N = 2: (#{&_2 + (&_2)_1}#){&_2 + &^&}#
N = 3: (#{&_2 + (&_2)_1}#){&_2 + &^&^&}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·2)
[95] #{&_2 + (&_2)_1}#>#
N = 1: #{&_2 + (&_2)_1}#
N = 2: (#{&_2 + (&_2)_1}#){&_2 + (&_2)_1}#
N = 3: ((#{&_2 + (&_2)_1}#){&_2 + (&_2)_1}#){&_2 + (&_2)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·ω)
[96] #{&_2 + (&_2)_1}##
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1}#>#{&_2 + (&_2)_1}#
N = 3: #{&_2 + (&_2)_1}#>#{&_2 + (&_2)_1}#>#{&_2 + (&_2)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·Ω)
[97] #{&_2 + (&_2)_1}#^#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1}##
N = 3: #{&_2 + (&_2)_1}###
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·Ω^ω)
[98] #{&_2 + (&_2)_1 + 1}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1}#{&_2 + (&_2)_1}#
N = 3: #{&_2 + (&_2)_1}#{&_2 + (&_2)_1}#{&_2 + (&_2)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·Ω^Ω)
[99] #{&_2 + (&_2)_1 + #}#
N = 1: #{&_2 + (&_2)_1 + 1}#
N = 2: #{&_2 + (&_2)_1 + 2}#
N = 3: #{&_2 + (&_2)_1 + 3}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·Ω^(Ω·ω))
[100] #{&_2 + (&_2)_1 + &}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1 + #{&_2 + (&_2)_1}#}#
N = 3: #{&_2 + (&_2)_1 + #{&_2 + (&_2)_1 + #{&_2 + (&_2)_1}#}#}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·Ω^Ω^2)
[101] #{&_2 + (&_2)_1 + &^#}#
N = 1: #{&_2 + (&_2)_1 + &}#
N = 2: #{&_2 + (&_2)_1 + &&}#
N = 3: #{&_2 + (&_2)_1 + &&&}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·Ω^Ω^ω)
[102] #{&_2 + (&_2)_1 + &^&}#
N = 1: #{&_2 + (&_2)_1 + &}#
N = 2: #{&_2 + (&_2)_1 + &#{&_2 + (&_2)_1 + &}#}#
N = 3: #{&_2 + (&_2)_1 + &^#{&_2 + (&_2)_1 + &^#{&_2 + (&_2)_1 + &}#}#}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)·Ω^Ω^Ω)
[103] #{&_2 + (&_2)_1 + (&_2)_1}#
N = 1: #{&_2 + (&_2)_1 + &}#
N = 2: #{&_2 + (&_2)_1 + &^&}#
N = 3: #{&_2 + (&_2)_1 + &^&^&}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)^2)
[104] #{&_2 + (&_2)_1*#}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1 + (&_2)_1}#
N = 3: #{&_2 + (&_2)_1 + (&_2)_1 + (&_2)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)^ω)
[105] #{&_2 + (&_2)_1*&}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1*#{&_2 + (&_2)_1}#}#
N = 3: #{&_2 + (&_2)_1*#{&_2 + (&_2)_1*#{&_2 + (&_2)_1}#}#}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)^Ω)
[106] #{&_2 + (&_2)_1*(&_2)_1}#
N = 1: #{&_2 + (&_2)_1*&}#
N = 2: #{&_2 + (&_2)_1*&^&}#
N = 3: #{&_2 + (&_2)_1*&^&^&}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)^ε(Ω + 1))
[107] #{&_2 + (&_2)_1^#}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1*(&_2)_1}#
N = 3: #{&_2 + (&_2)_1*(&_2)_1*(&_2)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)^ε(Ω + 1)^ω)
[108] #{&_2 + (&_2)_1^&}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1^#{&_2 + (&_2)_1}#}#
N = 3: #{&_2 + (&_2)_1^#{&_2 + (&_2)_1^#{&_2 + (&_2)_1}#}#}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)^ε(Ω + 1)^Ω)
[109] #{&_2 + (&_2)_1^(&_2)_1}#
N = 1: #{&_2 + (&_2)_1^&}#
N = 2: #{&_2 + (&_2)_1^&^&}#
N = 3: #{&_2 + (&_2)_1^&^&^&}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1))
[110] #{&_2 + (&_2 + 1)_1}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2)_1^(&_2)_1}#
N = 3: #{&_2 + (&_2)_1^(&_2)_1^(&_2)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + 2))
[111] #{&_2 + (&_2 + #)_1}#
N = 1: #{&_2 + (&_2 + 1)_1}#
N = 2: #{&_2 + (&_2 + 2)_1}#
N = 3: #{&_2 + (&_2 + 3)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω + ω))
[112] #{&_2 + (&_2 + &)_1}#
N = 1: #{&_2 + (&_2)_1}#
N = 2: #{&_2 + (&_2 + #{&_2 + (&_2)_1}#)_1}#
N = 3: #{&_2 + (&_2 + #{&_2 + (&_2 + #{&_2 + (&_2)_1}#)_1}#)_1}#
FGH ordinal level: ψ0(Ω_2·ε(Ω·2))
[113] #{&_2 + (&_2 + (&_2)_1)_1}#
N = 1: #{&_2 + (&_2 + &)_1}#
N = 2: #{&_2 + (&_2 + &^&)_1}#
N = 3: #{&_2 + (&_2 + &^&^&)_1}#
FGH ordinal level: ψ0(Ω_2·ε(ε(Ω + 1)))
And now, we are now at the halfway to &_3... Let's continue.
[114] #{&_2 + &_2}#
N = 1: #{&_2}#
N = 2: #{&_2 + (&_2)_1}#
N = 3: #{&_2 + (&_2 + (&_2)_1)_1}#
FGH ordinal level: ψ0(Ω_2^2)
But... It's way, way, way too difficult to define this that makes #{&2+&2}# standing at the ordinal level ψ0(Ω_2^2). Let's go to sneak peek number five!