Abandoned due to reformation. See current definition at the independent googology website here.
Here is the notation extension based on Saibian's Extended Cascading-E notation that goes up to the Bachmann-Howard ordinal, called Collapsing-E notation (E&).
Remember Sbiis ended Hyper-Extended Cascading-E Notation with a monstrous number:
(1) blasphemorgulus = E100#{&}#100
= E100#{#{#{...#{#{#}#}#...}#}#}#100 (with 100 #'s centered out, or 199 #'s on both sides (including the innermost one)).
He however defined {_,#,1,2} arrays only, along with {_,#+1,1,2} by his climbing theory.
However, he conclude that {_,#+2,1,2} doesn’t continue the nestings of {_,#+n,1,2} sequence in 4.3.12.
So, to continue. I have to think outside of the box! Remember the Extended Cascading-E Notation? Now to define the new symbol, & (ampersand), which diagonalizes over the transfinite recursion on #{α&}#.
From the Madore's theory, let the hyperion mark (#) to be analogous to the smallest transfinite ordinal ω, and the ampersand mark (&) to be analogous to the smallest uncountable ordinal Ω in the ordinal mathematics, using fundamental sequences for each limit of the Madore's psi function (ψ(α)).
Let the simplest expression of the Collapsing-E notation, #{&}#, to be defined as follows:
#{&}#[1] = #
#{&}#[n] = #{#{&}#[n-1]}#
And hence, #{&}#[2] = #{#}#, #{&}#[3] = #{#{#}#}#, and so on.
Let Ea#*a2#*...#*an be any expression in xE^, where a1 through an are n positive integers, and all #* are hyper-products (which may or may not be distinct.) Each individual #* may be chosen from the set of legal separators.
Then we have 5 formal rules of E&. Let #*k be the k-th hyper-product and L(#*k) be the last cascader of the k-th hyper-product:
Rule 1: Base rule: With no hyperions, we have Ea = 10^a.
Rule 2: Decomposition rule: If L(#*n-1) ≠ #^n (the last cascader is not of the form #^n, such as #, ##, #####, etc.):
E@a#*b = E@a#*[b]a (@ indicates the unchanged remainder of the expression and &[b] is the fundamental sequence of &).
Rule 3: Termination rule: If the last argument is 1, and the decomposition rule do not meet, it can be removed: E@a#*1 = E@a, for L(#*n) ≠ #^n.
Rule 4: Expansion rule: L(#*n-1) = #^n and #*n ≠ #: E@a(#*)*#b = E@a#*a(#*)*#b-1.
Rule 5: Recursion rule: Otherwise: E@a#b = E@(E@a#(b-1)).
In addition the set of legal delimiters must be defined. Let #* be the set of legal delimiters in xE^. The set is defined recursively:
In addition the set of legal delimiters must be defined. Let & be the set of legal delimiters in xE^. The set is defined recursively:
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)^^^...^^^^(b) w/n ^s, or (a){@+n}(b), where @ is the latter of the expression in {} and n is the finite number, is an element of #*
IV. If a,b are elements of #* and c is an element of #*+ , then (a)^^^...^^^(b)>(c) w/n ^s, or (a){@+n}(b), where @ is the latter of the expression in {} and n is the finite number, 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.
The decompositions of the pre-defined xE^ are defined as follows:
Case I. L= (α)^(β) where α,β ∈ &
A. When β = # :
IA1. &(α)^(#)[1] = &α
IA2. &(α)^(#)[n] = &α*(α)^(#)[n-1]
B. When β = ρ*# :
IB1. &(α)^(ρ*#)[1] = &(α)^(ρ)
IB2. &(α)^(ρ*#)[n] = &(α)^(ρ)*(α)^(ρ*#)[n-1]
C. When β ∈ &decomp:
IC1. &(α)^(β)[n] = &(α)^(β[n])
Case II. L= (α)^..k..^(β) where α,β ∈ & and k>1
A. When β = #:
IIA1. &(α)^..k..^(#)[1] = &α
IIA2. &(α)^..k..^(#)[n] = &(α)^..k-1..^((α)^..k..^(#)[n-1])
B. When β = ρ*#:
IIB1. &(α)^..k..^(ρ*#)[1] = &(α)^..k..^(ρ)
IIB2. &(α)^..k..^(ρ*#)[n] = &(α)^..k..^(ρ)>(α^..k..^(ρ*#)[n-1])
C. When β ∈ &decomp:
IIC1. &(α)^..k..^(β)[n] = &(α)^..k..^(β[n])
Case III. L= (α)^..k..^(β)>(γ) where α,β ∈& ,γ ∈ &+, and k>1
A. When γ = #:
IIIA1. &(α)^..k..^(β)>(#)[1] = &(α)^..k..^(β)
IIIA2. &(α)^..k..^(β)>(#)[n] = &((α)^..k..^(β)>(#)[n-1])^..k..^(β)
B. When γ=ρ+#:
IIIB1. &(α)^..k..^(β)>(ρ+#)[1] = &((α)^..k..^(β)>(ρ))^..k..^(β)
IIIB2. &(α)^..k..^(β)>(ρ+#)[n] = &((α)^..k..^(β)>(ρ+#)[n-1])^..k..^(β)
C. When γ ∈ &decomp:
IIIC1. (α)^..k..^(β)>(γ)[n] = (α)^..k..^(β)>(γ[n])
D. When γ=ρ+δ where ρ ∈ &+ and δ ∈ &decomp:
IIID1. (α)^..k..^(β)>(ρ+δ)[n] = (α)^..k..^(β)>(ρ+(δ[n]))
E. When γ=δ*# where δ ∈ &:
IIIE1. (α)^..k..^(β)>(δ*#)[1] = (α)^..k..^(β)>(δ)
IIIE2. (α)^..k..^(β)>(δ*#)[n] = (α)^..k..^(β)>(δ+δ*#)[n-1]
F. When γ =ρ+δ*# where ρ ∈ &+ and δ ∈ &:
IIIF1. (α)^..k..^(β)>(ρ+δ*#)[1] = (α)^..k..^(β)>(ρ+δ)
IIIF2. (α)^..k..^(β)>(ρ+δ*#)[n] = (α)^..k..^(β)>(ρ+δ+δ*#)[n-1]
And in #xE^:
Rule I. When γ = #: &(α){#}#[n] = &(α)^..n..^#
Rule II. When γ = ρ+#: &(α){ρ+#}#[n] = &(α){ρ+n}#
Rule III. When γ ∈ &decomp: &(α){γ}#[n] = &(α){γ[n]}#
Rule IV. When γ = ρ+δ, ρ ∈ &+, δ ∈ &decomp: &(α){ρ+δ}#[n] = &(α){ρ+(δ[n])}#
Rule V. When γ = δ*# where δ ∈ &+:
V1. &(α){δ*#}#[1] = &(α){δ}#
V2. &(α){δ*#}#[n] = &(α){δ+δ*#}#[n-1]
Rule VI. When γ = ρ+δ*#, ρ ∈ &+, δ ∈ &:
VI1. &(α){ρ+δ*#}#[1] = &(α){ρ+δ}#
VI2. &(α){ρ+δ*#}#[n] = &(α){ρ+δ+δ*#}#[n-1]
Rule VII. When γ is a successor ordinal, consult the rules for xE^.
To define the formal rules for the "&" delimiter, we have to define the rule as follows:
Rule VIII. When γ = &:
VII1. #*(α){&}#[1] = #*(α)#
VII2. #*(α){&}#[n] = #*(α){#*(α){&}#[n-1]}#
Rule IX. When γ = ρ+&:
IX1. #*(α){ρ+&}#[1] = #*(α){ρ+#}#
IX2. #*(α){ρ+&}#[n] = #*(α){ρ+#*(α){ρ+&}#[n-1]}#
Rule X. When γ = ρ*&:
X1. #*(α){ρ*&}#[1] = #*(α){ρ*#}#
X2. #*(α){ρ*&}#[n] = #*(α){ρ*#*(α){ρ*&}#[n-1]}#
Rule XI. When γ = ρ^&:
XI1. #*(α){ρ^&}#[1] = #*(α){ρ^#}#
XI2. #*(α){ρ^&}#[n] = #*(α){ρ^#*(α){ρ^&}#[n-1]}#
Thus, the extension of the notation can now go up to f_ψ(ε(Ω+1))(n), which is the Bachmann-Howard ordinal level.
Examples:
E100#{&}#4
= E100#{&}#[4]100
= E100#{#{&}#[3]}#100
= E100#{#{#{&}#[2]}#}#100
= E100#{#{#{#}#}#}#100
= E100#{#{#{100}#}#}#100
E100#{&&+&}#3
= E100#{&&+&}#[3]100
= E100#{&&+#{&&+&}#[2]}#100
= E100#{&&+#{&&+#{&&+&}#[1]}#}#100
= E100#{&&+#{&&+#{&&+#}#}#}#100
= E100#{&&+#{&&+#{&&+100}#}#}#100
E100#{&^&^#}#5
= E100#{&^&^#}#[5]100
= E100#{&^&^#[5]}#100
= E100#{&^(&^#[5])}#100
= E100#{&^(&*&^#[4])}#100
= E100#{&^(&&*&^#[3])}#100
= ...
= E100#{&^&&&&&}#100
= E100#{&^&&&&&}#[100]100
= E100#{&^(&&&&*#{&^&&&&&}#[99])}#100
= E100#{&^(&&&&*#{&^&&&&*#{&^&&&&&}#[98]}#)}#100
= ...
E100#{&^&*&&}#3
= E100#{&^&*&&}#[3]100
= E100#{&^&*&*#{&^&*&&}#[2]}#100
= E100#{&^&*&*#{&^&*&*#{&^&*&&}#[1]}#}#100
= E100#{&^&*&*#{&^&*&*#{&^&*&#}#}#}#100
E100(#{&}#){&}#4
= E100(#{&}#){&}#[4]100
= E100(#{&}#){(#{&}#){&}#[3]}#100
= E100(#{&}#){(#{&}#){(#{&}#){&}#[2]}#}#100
= E100(#{&}#){(#{&}#){(#{&}#){(#{&}#){&}#[1]}#}#}#100
= E100(#{&}#){(#{&}#){(#{&}#){#{&}#}#}#}#100
Here ψ is the extended Buchholz's function, and using system of fundamental sequences based on the extended Buchholz's function.
#{&}# has level φ(1,0,0,0) = ψ(Ω^Ω^2)
#{&}#*# has level φ(1,0,0,0)*ω = ψ(Ω^Ω^2+1)
#{&}#*#{&}# has level φ(1,0,0,0)^2 = ψ(Ω^Ω^2+ψ(Ω^Ω^2))
(#{&}#)^# has level φ(1,0,0,0)^ω = ψ(Ω^Ω^2+ψ(Ω^Ω^2+1))
(#{&}#)^(#{&}#) has level φ(1,0,0,0)^φ(1,0,0,0) = ψ(Ω^Ω^2+ψ(Ω^Ω^2+ψ(Ω^Ω^2)))
(#{&}#)^^# has level ε(φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω)
((#{&}#)^^#)^^# has level ε(φ(1,0,0,0)+2) = ψ(Ω^Ω^2+Ω2)
(#{&}#)^^#># has level ε(φ(1,0,0,0)+ω) = ψ(Ω^Ω^2+Ωω)
(#{&}#)^^#>(#{&}#) has level ε(φ(1,0,0,0)*2) = ψ(Ω^Ω^2+Ω*ψ(Ω^Ω^2))
(#{&}#)^^#>(#{&}#)^^# has level ε(ε(φ(1,0,0,0)+1)) = ψ(Ω^Ω^2+Ω*ψ(Ω^Ω^2+1))
(#{&}#)^^## has level ζ(φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^2)
(#{&}#)^^### has level η(φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^3)
(#{&}#)^^#### has level φ(4,φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^4)
(#{&}#)^^#^# has level φ(ω,φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^ω)
(#{&}#)^^#^^# has level φ(ε0,φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^ψ(Ω))
(#{&}#)^^(#{&}#) has level φ(φ(1,0,0,0),1) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2))
((#{&}#)^^(#{&}#))^^(#{&}#) has level φ(φ(1,0,0,0),2) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2)*2)
(#{&}#)^^(#{&}#)># has level φ(φ(1,0,0,0),ω) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2)*ω)
(#{&}#)^^(#{&}#)>(#{&}#) has level φ(φ(1,0,0,0),φ(1,0,0,0)) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2)*ψ(Ω^Ω^2))
(#{&}#)^^(#{&}#)>(#{&}#)^^(#{&}#) has level φ(φ(1,0,0,0),φ(φ(1,0,0,0),1)) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2)*ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2)))
(#{&}#)^^(#{&}#*#) has level φ(φ(1,0,0,0)+1,0) = ψ(Ω^Ω^2+Ω^(ψ(Ω^Ω^2)+1))
(#{&}#)^^(#{&}#*#{&}#) has level φ(φ(1,0,0,0)*2,0) = ψ(Ω^Ω^2+Ω^(ψ(Ω^Ω^2)*2))
(#{&}#)^^(#{&}#)^# has level φ(φ(1,0,0,0)*ω,0) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2+1))
(#{&}#)^^(#{&}#)^^# has level φ(ε(φ(1,0,0,0)+1),0) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2+Ω))
(#{&}#)^^(#{&}#)^^(#{&}#) has level φ(φ(φ(1,0,0,0),1),0) = ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2+Ω^ψ(Ω^Ω^2)))
(#{&}#)^^^# has level Γ(φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^Ω)
(#{&}#)^^^^# has level φ(2,0,φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^Ω2)
(#{&}#){#}# has level φ(ω,0,φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^Ωω)
(#{&}#){#{#}#}# has level φ(φ(ω,0,0),0,φ(1,0,0,0)+1) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ωω)))
(#{&}#){#{&}#}# has level φ(φ(1,0,0,0),0,1) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)))
((#{&}#){#{&}#}#){#{&}#}# has level φ(φ(1,0,0,0),0,2) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2))2)
(#{&}#){#{&}#}#># has level φ(φ(1,0,0,0),0,ω) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2))ω)
(#{&}#){#{&}#}#>(#{&}#) has level φ(φ(1,0,0,0),0,φ(1,0,0,0)) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2))*ψ(Ω^Ω^2))
(#{&}#){#{&}#}#>(#{&}#){#{&}#}# has level φ(φ(1,0,0,0),0,φ(φ(1,0,0,0),0,1)) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2))*ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2))))
(#{&}#){#{&}#}## has level φ(φ(1,0,0,0),1,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)+1))
(#{&}#){#{&}#}#^# has level φ(φ(1,0,0,0),ω,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)+ω))
(#{&}#){#{&}#}(#{&}#) has level φ(φ(1,0,0,0),φ(1,0,0,0),0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)+ψ(Ω^Ω^2)))
(#{&}#){#{&}#+1}# has level φ(φ(1,0,0,0)+1,0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)+Ω))
(#{&}#){#{&}#+#}# has level φ(φ(1,0,0,0)+ω,0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)+Ωω))
(#{&}#){#{&}#+#{&}#}# has level φ(φ(1,0,0,0)2,0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)2))
(#{&}#){#{&}#*#}# has level φ(φ(1,0,0,0)ω,0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2+1)))
(#{&}#){#{&}#*#{&}#}# has level φ(φ(1,0,0,0)^2,0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2+ψ(Ω^Ω^2))))
(#{&}#){(#{&}#)^#}# has level φ(φ(1,0,0,0)^ω,0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2+ψ(Ω^Ω^2+1))))
(#{&}#){(#{&}#)^^#}# has level φ(ε(φ(1,0,0,0)+1),0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2+Ω)))
(#{&}#){(#{&}#)^^^#}# has level φ(Γ(φ(1,0,0,0)+1),0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2+Ω^Ω)))
(#{&}#){(#{&}#){#}#}# has level φ(φ(ω,0,φ(1,0,0,0)+1),0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2+Ω^Ωω)))
(#{&}#){(#{&}#){#{&}#}#}# has level φ(φ(φ(1,0,0,0),0,1),0,0) = ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2+Ω^(Ω*ψ(Ω^Ω^2)))))
...
(#{&}#){&}# has level φ(1,0,0,1) = ψ(Ω^Ω^2*2)
((#{&}#){&}#){#{&}#}# has level φ(φ(1,0,0,0),0,φ(1,0,0,1)+1) = ψ(Ω^Ω^2*2+Ω^(Ω*ψ(Ω^Ω^2)))
((#{&}#){&}#){(#{&}#){&}#}# has level φ(φ(1,0,0,1),0,1) = ψ(Ω^Ω^2*2+Ω^(Ω*ψ(Ω^Ω^2*2)))
((#{&}#){&}#){&}# has level φ(1,0,0,2) = ψ(Ω^Ω^2*3)
#{&}#># has level φ(1,0,0,ω) = ψ(Ω^Ω^2*ω)
(#{&}#>#){&}# has level φ(1,0,0,ω+1) = ψ(Ω^Ω^2*ω+Ω^Ω^2)
#{&}#>(#+#) has level φ(1,0,0,ω2) = ψ(Ω^Ω^2*ω2)
#{&}#>## has level φ(1,0,0,ω^2) = ψ(Ω^Ω^2*ω^2)
#{&}#>#^# has level φ(1,0,0,ω^ω) = ψ(Ω^Ω^2*ω^ω)
#{&}#>#^^# has level φ(1,0,0,ε0) = ψ(Ω^Ω^2*ψ(Ω))
#{&}#>#^^## has level φ(1,0,0,ζ0) = ψ(Ω^Ω^2*ψ(Ω^2))
#{&}#>#^^^# has level φ(1,0,0,Γ0) = ψ(Ω^Ω^2*ψ(Ω^Ω))
#{&}#>#{&}# has level φ(1,0,0,φ(1,0,0,0)) = ψ(Ω^Ω^2*ψ(Ω^Ω^2))
#{&}#>#{&}#># has level φ(1,0,0,φ(1,0,0,0)) = ψ(Ω^Ω^2*ψ(Ω^Ω^2*ω))
#{&}#>#{&}#>#{&}# has level φ(1,0,0,φ(1,0,0,φ(1,0,0,0))) = ψ(Ω^Ω^2*ψ(Ω^Ω^2*ψ(Ω^Ω^2)))
...
#{&}## has level φ(1,0,1,0) = ψ(Ω^(Ω^2+1))
(#{&}##){#{&}#}# has level φ(φ(1,0,0,0),0,φ(1,0,1,0)+1) = ψ(Ω^(Ω^2+1)+Ω^(Ω*ψ(Ω^Ω^2)))
(#{&}##){#{&}##}# has level φ(φ(1,0,1,0),0,1) = ψ(Ω^(Ω^2+1)+Ω^(Ω*ψ(Ω^(Ω^2+1))))
(#{&}##){&}# has level φ(1,0,0,φ(1,0,1,0)+1) = ψ(Ω^(Ω^2+1)+Ω^Ω^2)
((#{&}##){&}#){&}# has level φ(1,0,0,φ(1,0,1,0)+2) = ψ(Ω^(Ω^2+1)+Ω^Ω^2*2)
(#{&}##){&}#># has level φ(1,0,0,φ(1,0,1,0)+ω) = ψ(Ω^(Ω^2+1)+Ω^Ω^2*ω)
(#{&}##){&}#>(#{&}##){&}# has level φ(1,0,0,φ(1,0,0,φ(1,0,1,0)+1)) = ψ(Ω^(Ω^2+1)+Ω^Ω^2*ψ(Ω^(Ω^2+1)))
(#{&}##){&}## has level φ(1,0,1,1) = ψ(Ω^(Ω^2+1)*2)
#{&}##># has level φ(1,0,1,ω) = ψ(Ω^(Ω^2+1)*ω)
#{&}##>#{&}## has level φ(1,0,1,φ(1,0,1,0)) = ψ(Ω^(Ω^2+1)*ψ(Ω^(Ω^2+1)))
#{&}### has level φ(1,0,2,0) = ψ(Ω^(Ω^2+2))
#{&}#### has level φ(1,0,3,0) = ψ(Ω^(Ω^2+3))
#{&}#^# has level φ(1,0,ω,0) = ψ(Ω^(Ω^2+ω))
#{&}#^^# has level φ(1,0,ε0,0) = ψ(Ω^(Ω^2+ψ(Ω)))
#{&}#^^## has level φ(1,0,ζ0,0) = ψ(Ω^(Ω^2+ψ(Ω^2)))
#{&}#^^^# has level φ(1,0,Γ0,0) = ψ(Ω^(Ω^2+ψ(Ω^Ω)))
#{&}#{&}# has level φ(1,0,φ(1,0,0,0),0) = ψ(Ω^(Ω^2+ψ(Ω^Ω^2)))
#{&}#{&}#{&}# has level φ(1,0,φ(1,0,φ(1,0,0,0),0),0) = ψ(Ω^(Ω^2+ψ(Ω^(Ω^2+ψ(Ω^Ω^2)))))
...
#{&+1}# has level φ(1,1,0,0) = ψ(Ω^(Ω^2+Ω))
(#{&+1}#)^^# has level ε(φ(1,1,0,0)+1) = ψ(Ω^(Ω^2+Ω)+Ω)
(#{&+1}#){#{&}#}# has level φ(φ(1,0,0,0),0,φ(1,1,0,0)+1) = ψ(Ω^(Ω^2+Ω)+Ω^(Ω*ψ(Ω^Ω^2)))
(#{&+1}#){#{&+1}#}# has level φ(φ(1,1,0,0),0,1) = ψ(Ω^(Ω^2+Ω)+Ω^(Ω*ψ(Ω^(Ω^2+Ω))))
(#{&+1}#){&}# has level φ(1,0,0,φ(1,1,0,0)+1) = ψ(Ω^(Ω^2+Ω)+Ω^Ω^2)
(#{&+1}#){&}#># has level φ(1,0,0,φ(1,1,0,0)+ω) = ψ(Ω^(Ω^2+Ω)+Ω^Ω^2*ω)
(#{&+1}#){&}#>(#{&+1}#){&}# has level φ(1,0,0,φ(1,0,0,φ(1,1,0,0)+1)) = ψ(Ω^(Ω^2+Ω)+Ω^Ω^2*ψ(Ω^(Ω^2+Ω)))
(#{&+1}#){&}## has level φ(1,0,1,φ(1,1,0,0)+1) = ψ(Ω^(Ω^2+Ω)+Ω^(Ω^2+1))
(#{&+1}#){&}#^# has level φ(1,0,ω,φ(1,1,0,0)+1) = ψ(Ω^(Ω^2+Ω)+Ω^(Ω^2+ω))
(#{&+1}#){&}#{&}# has level φ(1,0,φ(1,0,0,0),φ(1,1,0,0)+1) = ψ(Ω^(Ω^2+Ω)+Ω^(Ω^2+ψ(Ω^Ω^2)))
(#{&+1}#){&}(#{&+1}#) has level φ(1,0,φ(1,1,0,0),1) = ψ(Ω^(Ω^2+Ω)+Ω^(Ω^2+ψ(Ω^(Ω^2+1))))
(#{&+1}#){&+1}# has level φ(1,1,0,1) = ψ(Ω^(Ω^2+Ω)*2)
#{&+1}#># has level φ(1,1,0,ω) = ψ(Ω^(Ω^2+Ω)*ω)
#{&+1}#>#{&+1}# has level φ(1,1,0,φ(1,1,0,0)) = ψ(Ω^(Ω^2+Ω)*ψ(Ω^(Ω^2+Ω)))
#{&+1}## has level φ(1,1,1,0) = ψ(Ω^(Ω^2+Ω+1))
#{&+1}#^# has level φ(1,1,ω,0) = ψ(Ω^(Ω^2+Ω+ω))
#{&+1}#{&+1}# has level φ(1,1,φ(1,1,0,0),0) = ψ(Ω^(Ω^2+Ω+ψ(Ω^(Ω^2+Ω))))
#{&+2}# has level φ(1,2,0,0) = ψ(Ω^(Ω^2+Ω2))
#{&+3}# has level φ(1,3,0,0) = ψ(Ω^(Ω^2+Ω3))
#{&+#}# has level φ(1,ω,0,0) = ψ(Ω^(Ω^2+Ωω))
#{&+#^^#}# has level φ(1,ε0,0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω)))
#{&+#^^##}# has level φ(1,ζ0,0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^2)))
#{&+#^^^#}# has level φ(1,Γ0,0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^Ω)))
#{&+#{&}#}# has level φ(1,φ(1,0,0,0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^Ω^2)))
#{&+(#{&}#){&}#}# has level φ(1,φ(1,0,0,1),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^Ω^2*2)))
#{&+#{&}#>#}# has level φ(1,φ(1,0,0,ω),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^Ω^2*ω)))
#{&+#{&}#>#{&}#}# has level φ(1,φ(1,0,0,φ(1,0,0,0)),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^Ω^2*ψ(Ω^Ω^2))))
#{&+#{&}##}# has level φ(1,φ(1,0,1,0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+1))))
#{&+#{&}#^#}# has level φ(1,φ(1,0,ω,0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+ω))))
#{&+#{&}#{&}#}# has level φ(1,φ(1,0,φ(1,0,0,0),0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+ψ(Ω^Ω^2)))))
#{&+#{&+1}#}# has level φ(1,φ(1,1,0,0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+Ω))))
#{&+#{&+#}#}# has level φ(1,φ(1,ω,0,0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+Ωω))))
#{&+#{&+#{&}#}#}# has level φ(1,φ(1,φ(1,0,0,0),0,0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+Ω*ψ(Ω^Ω^2)))))
#{&+#{&+#{&+#}#}#}# has level φ(1,φ(1,φ(1,ω,0,0),0,0),0,0) = ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+Ω*ψ(Ω^(Ω^2+Ωω))))))
...
#{&+&}# has level φ(2,0,0,0) = ψ(Ω^(Ω^2*2))
(#{&+&}#){&}# has level φ(1,0,0,φ(2,0,0,0)+1) = ψ(Ω^(Ω^2*2)+Ω^Ω^2)
(#{&+&}#){&+&}# has level φ(2,0,0,1) = ψ(Ω^(Ω^2*2)*2)
#{&+&}#># has level φ(2,0,0,ω) = ψ(Ω^(Ω^2*2)*ω)
#{&+&}#{&+&}# has level φ(2,0,0,φ(2,0,0,0)) = ψ(Ω^(Ω^2*2)*ψ(Ω^(Ω^2*2)))
#{&+&}## has level φ(2,0,1,0) = ψ(Ω^(Ω^2*2+1))
#{&+&}#^# has level φ(2,0,ω,0) = ψ(Ω^(Ω^2*2+ω))
#{&+&}#{&+&}# has level φ(2,0,φ(2,0,0,0),0) = ψ(Ω^(Ω^2*2+ψ(Ω^(Ω^2*2))))
#{&+&+1}# has level φ(2,1,0,0) = ψ(Ω^(Ω^2*2+Ω))
#{&+&+#}# has level φ(2,ω,0,0) = ψ(Ω^(Ω^2*2+Ωω))
#{&+&+#{&+&}#}# has level φ(2,φ(2,0,0,0),0,0) = ψ(Ω^(Ω^2*2+Ωψ(Ω^(Ω^2*2))))
#{&+&+#{&+&+#}#}# has level φ(2,φ(2,ω,0,0),0,0) = ψ(Ω^(Ω^2*2+Ωψ(Ω^(Ω^2*2+Ωω))))
#{&+&+&}# has level φ(3,0,0,0) = ψ(Ω^(Ω^2*3))
#{&+&+&+&}# has level φ(4,0,0,0) = ψ(Ω^(Ω^2*4))
...
#{&*#}# has level φ(ω,0,0,0) = ψ(Ω^(Ω^2*ω))
(#{&*#}#){&}# has level φ(1,0,0,φ(ω,0,0,0)+1) = ψ(Ω^(Ω^2*ω)+Ω^Ω^2)
(#{&*#}#){&*2}# has level φ(ω,0,0,1) = ψ(Ω^(Ω^2*ω)*2)
#{&*#}#># has level φ(ω,0,0,ω) = ψ(Ω^(Ω^2*ω)*ω)
#{&*#}## has level φ(ω,0,1,0) = ψ(Ω^(Ω^2*ω+1))
#{&*#}#^# has level φ(ω,0,ω,0) = ψ(Ω^(Ω^2*ω+ω))
#{&*#+1}# has level φ(ω,1,0,0) = ψ(Ω^(Ω^2*ω+Ω))
#{&*#+#}# has level φ(ω,ω,0,0) = ψ(Ω^(Ω^2*ω+Ωω))
#{&*#+#{&*#}#}# has level φ(ω,φ(ω,0,0,0),0,0) = ψ(Ω^(Ω^2*ω+Ω*ψ(Ω^(Ω^2*ω))))
#{&*#+&}# has level φ(ω+1,0,0,0) = ψ(Ω^(Ω^2*ω+Ω^2))
#{&*#+&+&}# has level φ(ω+2,0,0,0) = ψ(Ω^(Ω^2*ω+Ω^2*2))
#{&*#+&*#}# has level φ(ω2,0,0,0) = ψ(Ω^(Ω^2*ω2))
#{&*##}# has level φ(ω^2,0,0,0) = ψ(Ω^(Ω^2*ω^2))
#{&*#^#}# has level φ(ω^ω,0,0,0) = ψ(Ω^(Ω^2*ω^ω))
#{&*#^^#}# has level φ(ε0,0,0,0) = ψ(Ω^(Ω^2*ψ(Ω)))
#{&*#^^##}# has level φ(ζ0,0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^2)))
#{&*#^^^#}# has level φ(Γ0,0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^Ω)))
#{&*#{&}#}# has level φ(φ(1,0,0,0),0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^Ω^2)))
#{&*#{&+1}#}# has level φ(φ(1,1,0,0),0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^(Ω^2+1))))
#{&*#{&+&}#}# has level φ(φ(2,0,0,0),0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^(Ω^2*2))))
#{&*#{&*#}#}# has level φ(φ(ω,0,0,0),0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^(Ω^2*ω))))
#{&*#{&*#{&}#}#}# has level φ(φ(φ(1,0,0,0),0,0,0),0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^(Ω^2*ψ(Ω^Ω^2)))))
#{&*#{&*#{&*#}#}#}# has level φ(φ(φ(ω,0,0,0),0,0,0),0,0,0) = ψ(Ω^(Ω^2*ψ(Ω^(Ω^2*ψ(Ω^(Ω^2*ω))))))
...
#{&&}# has level φ(1,0,0,0,0) = ψ(Ω^Ω^3)
(#{&&}#){&}# has level φ(1,0,0,φ(1,0,0,0,0)+1) = ψ(Ω^Ω^3+Ω^Ω^2)
(#{&&}#){&&}# has level φ(1,0,0,0,1) = ψ(Ω^Ω^3*2)
#{&&}#># has level φ(1,0,0,0,ω) = ψ(Ω^Ω^3*ω)
#{&&}#>#{&&}# has level φ(1,0,0,0,φ(1,0,0,0,0)) = ψ(Ω^Ω^3*ψ(Ω^Ω^3))
#{&&}## has level φ(1,0,0,1,0) = ψ(Ω^(Ω^3+1))
#{&&}#^# has level φ(1,0,0,ω,0) = ψ(Ω^(Ω^3+ω))
#{&&}#{&&}# has level φ(1,0,0,φ(1,0,0,0,0),0) = ψ(Ω^(Ω^3+ψ(Ω^Ω^3)))
#{&&+1}# has level φ(1,0,1,0,0) = ψ(Ω^(Ω^3+Ω))
#{&&+#}# has level φ(1,0,ω,0,0) = ψ(Ω^(Ω^3+Ωω))
#{&&+#{&&}#}# has level φ(1,0,φ(1,0,0,0,0),0,0) = ψ(Ω^(Ω^3+Ω*ψ(Ω^Ω^3)))
#{&&+&}# has level φ(1,1,0,0,0) = ψ(Ω^(Ω^3+Ω^2))
#{&&+&*#}# has level φ(1,ω,0,0,0) = ψ(Ω^(Ω^3+Ω^2*ω))
#{&&+&*#{&&}#}# has level φ(1,φ(1,0,0,0,0),0,0,0) = ψ(Ω^(Ω^3+Ω^2*ψ(Ω^Ω^3)))
#{&&+&&}# has level φ(2,0,0,0,0) = ψ(Ω^(Ω^3*2))
#{&&*#}# has level φ(ω,0,0,0,0) = ψ(Ω^(Ω^3*ω))
#{&&*#{&&}#}# has level φ(φ(1,0,0,0,0),0,0,0,0) = ψ(Ω^(Ω^3*ψ(Ω^Ω^3)))
#{&&*#{&&*#}#}# has level φ(φ(ω,0,0,0,0),0,0,0,0) = ψ(Ω^(Ω^3*ψ(Ω^(Ω^3*ω))))
#{&&&}# has level φ(1,0,0,0,0,0) = ψ(Ω^Ω^4)
#{&&&+&&&}# has level φ(2,0,0,0,0,0) = ψ(Ω^(Ω^4*2))
#{&&&*#}# has level φ(ω,0,0,0,0,0) = ψ(Ω^(Ω^4*ω))
#{&&&&}# has level φ(1,0,0,0,0,0,0) = ψ(Ω^Ω^5)
#{&&&&&}# has level φ(1,0,0,0,0,0,0,0) = ψ(Ω^Ω^6)
...
#{&^#}# has level ψ(Ω^Ω^ω) (small Veblen ordinal, SVO)
(#{&^#}#)^^# has level ψ(Ω^Ω^ω+Ω)
(#{&^#}#)^^## has level ψ(Ω^Ω^ω+Ω^2)
(#{&^#}#)^^^# has level ψ(Ω^Ω^ω+Ω^Ω)
(#{&^#}#){&}# has level ψ(Ω^Ω^ω+Ω^Ω^2)
(#{&^#}#){&^#}# has level ψ(Ω^Ω^ω*2)
#{&^#}#># has level ψ(Ω^Ω^ω*ω)
#{&^#}#>#{&^#}# has level ψ(Ω^Ω^ω*ψ(Ω^Ω^ω))
#{&^#}#>#{&^#}# has level ψ(Ω^Ω^ω*ψ(Ω^Ω^ω))
#{&^#}## has level ψ(Ω^(Ω^ω+1))
#{&^#}#^# has level ψ(Ω^(Ω^ω+ω))
#{&^#}#{&^#}# has level ψ(Ω^(Ω^ω+ψ(Ω^Ω^ω)))
#{&^#+1}# has level ψ(Ω^(Ω^ω+Ω))
#{&^#+#}# has level ψ(Ω^(Ω^ω+Ωω))
#{&^#+#{&^#}#}# has level ψ(Ω^(Ω^ω+Ωψ(Ω^Ω^ω)))
#{&^#+&}# has level ψ(Ω^(Ω^ω+Ω^2))
#{&^#+&&}# has level ψ(Ω^(Ω^ω+Ω^3))
#{&^#+&^#}# has level ψ(Ω^(Ω^ω*2))
#{&^#*#}# has level ψ(Ω^(Ω^ω*ω))
#{&^#*#{&^#}#}# has level ψ(Ω^(Ω^ω*ψ(Ω^Ω^ω)))
#{&^#*&}# has level ψ(Ω^Ω^(ω+1))
#{&^#*&&}# has level ψ(Ω^Ω^(ω+2))
#{&^#*&^#}# has level ψ(Ω^Ω^ω2)
#{&^##}# has level ψ(Ω^Ω^ω^2)
#{&^#^#}# has level ψ(Ω^Ω^ω^ω)
#{&^#^^#}# has level ψ(Ω^Ω^ψ(Ω))
#{&^#^^##}# has level ψ(Ω^Ω^ψ(Ω^2))
#{&^#^^^#}# has level ψ(Ω^Ω^ψ(Ω^Ω))
#{&^#{&}#}# has level ψ(Ω^Ω^ψ(Ω^Ω^2))
#{&^#{&&}#}# has level ψ(Ω^Ω^ψ(Ω^Ω^3))
#{&^#{&^#}#}# has level ψ(Ω^Ω^ψ(Ω^Ω^ω))
#{&^#{&^#{&^#}#}#}# has level ψ(Ω^Ω^ψ(Ω^Ω^ψ(Ω^Ω^ω)))
...
#{&^&}# has level ψ(Ω^Ω^Ω) (large Veblen ordinal, LVO)
(#{&^&}#)^^# has level ψ(Ω^Ω^Ω+Ω)
(#{&^&}#)^^## has level ψ(Ω^Ω^Ω+Ω^2)
(#{&^&}#)^^^# has level ψ(Ω^Ω^Ω+Ω^Ω)
(#{&^&}#){&}# has level ψ(Ω^Ω^Ω+Ω^Ω^2)
(#{&^&}#){&^#}# has level ψ(Ω^Ω^Ω+Ω^Ω^ω)
(#{&^&}#){&^&}# has level ψ(Ω^Ω^Ω*2)
#{&^&}#># has level ψ(Ω^Ω^Ω*ω)
#{&^&}#>#{&^&}# has level ψ(Ω^Ω^Ω*ψ(Ω^Ω^Ω))
#{&^&+1}# has level ψ(Ω^(Ω^Ω+1))
#{&^&+#}# has level ψ(Ω^(Ω^Ω+ω))
#{&^&+#{&^&}#}# has level ψ(Ω^(Ω^Ω+ψ(Ω^Ω^Ω)))
#{&^&+&}# has level ψ(Ω^(Ω^Ω+Ω))
#{&^&+&&}# has level ψ(Ω^(Ω^Ω+Ω^2))
#{&^&+&^#}# has level ψ(Ω^(Ω^Ω+Ω^ω))
#{&^&+&^&}# has level ψ(Ω^(Ω^Ω*2))
#{&^&*#}# has level ψ(Ω^(Ω^Ω*ω))
#{&^&*#{&^&}#}# has level ψ(Ω^(Ω^Ω*ψ(Ω^Ω^Ω)))
#{&^&*&}# has level ψ(Ω^Ω^(Ω+1))
#{&^&*&&}# has level ψ(Ω^Ω^(Ω+2))
#{&^&*&^#}# has level ψ(Ω^Ω^(Ω+ω))
#{&^&*&^#{&^&}#}# has level ψ(Ω^Ω^(Ω+ψ(Ω^Ω^Ω)))
#{&^&*&^&}# has level ψ(Ω^Ω^Ω2)
#{&^(&*#)}# has level ψ(Ω^Ω^Ωω)
#{&^(&*#{&^&}#)}# has level ψ(Ω^Ω^(Ω*ψ(Ω^Ω^Ω)))
#{&^&&}# has level ψ(Ω^Ω^Ω^2)
#{&^&&&}# has level ψ(Ω^Ω^Ω^3)
#{&^&^#}# has level ψ(Ω^Ω^Ω^ω)
#{&^&^#{&^&}#}# has level ψ(Ω^Ω^Ω^ψ(Ω^Ω^Ω))
#{&^&^&}# has level ψ(Ω^Ω^Ω^Ω)
#{&^&^&+1}# has level ψ(Ω^(Ω^Ω^Ω+1))
#{&^&^&+&^&^&}# has level ψ(Ω^(Ω^Ω^Ω*2))
#{&^&^&*#}# has level ψ(Ω^(Ω^Ω^Ω*ω))
#{&^&^&*&}# has level ψ(Ω^Ω^(Ω^Ω+1))
#{&^&^&*&^&^&}# has level ψ(Ω^Ω^(Ω^Ω*2))
#{&^(&^&*#)}# has level ψ(Ω^Ω^(Ω^Ω*ω))
#{&^(&^&*&)}# has level ψ(Ω^Ω^Ω^(Ω+1))
#{&^(&^&*&^&)}# has level ψ(Ω^Ω^Ω^Ω2)
#{&^&^(&*#)}# has level ψ(Ω^Ω^Ω^Ωω)
#{&^&^&&}# has level ψ(Ω^Ω^Ω^Ω^2)
#{&^&^&^#}# has level ψ(Ω^Ω^Ω^Ω^ω)
#{&^&^&^&}# has level ψ(Ω^Ω^Ω^Ω^Ω)
#{&^&^&^&^&}# has level ψ(Ω^Ω^Ω^Ω^Ω^Ω)
Now, the limit is the Bachmann-Howard ordinal (BHO, ψ(ε(Ω+1)) = ψ(Ω^Ω^Ω^...)).