Following the nearly 3-month hiatus for creating the Collapsing-E notation from the previous sneak peak, I decided to explain the intermediate level of the Collapsing-E notation from #{&}# to (#{&}#){&}#!
We adopt the "ampersand" symbol (&) as the least hyper-delimiter that the symbol is devised from the CompactStar's ampersand notation. This hyper-delimiter diagonalizes over the previous recursion level in which the strength is comparable to the first uncountable cardinal of the ordinal collapsing functions. However, the "&" delimiter itself has no fundamental sequences, hence it must be used inside {} via the decomposition rules.
We decide to choose the extended (finitary and transfinitary) Veblen function that extends to the large Veblen ordinal, since the limit of the Hyper-Extended Cascading-E notation (using extended Veblen hierarchy) is φ(1,0,0,0) in the fast-growing hierarchy.
First, the limit of the (#{&}#){#{#{...#{#{#}#}#...}#}#}# is not (#{&}#){&}#, but (#{&}#){#{&}#}#. This because the rule on "&" hyper-delimiter for the &E system is created recursively by:
Let E a1 %(1) a2 %(2) a3 %(3) ... a[n-1] %(n-1) an
where a1 ~ an, and n are natural numbers (positive integers)
and %(1) ~ %(k-1) are k-1 delimiters from the set x^
and k ∈ {1, 2, 3, 4, 5, 6, ...} be defined as follows: Let
m denote ak-1
n denote ak
@ denote the unchanged remainder of an expression
% denote any portion of a delimiter we chose to omit
and %[n] denote the nth member of the fundamental sequence defined for the delimiter %
Furthermore, the delimiter (#{&}#){&}# is expected to have the FGH ordinal level of φ(1,0,0,1), while (#{&}#){#{&}#}#'s φ(φ(1,0,0,0),0,1), this because φ(1,0,0,1) ≠ φ(φ(1,0,0,0),0,1) using the Veblen hierarchy.
With the new rules in definition of %[n]:
A. For the #{&}# delimiter: #{&}#[1] = #, #{&}#[n+1] = #{#{&}#[n]}#
B. When b = & in {}:
B.1. %(a){&}#[0] = #
B.2. %(a){&}#[1] = %(a)
B.3. %(a){&}#[2] = %(a){a}#
B.4. %(a){&}#[n+1] = %(a){(a){&}#[n]}# for n > 2
For example:
#{&}#[0] = #{&}#[1] = #
(#{&}#){&}#[1] = #{&}#
etc.
With non-trivial cases:
#{&}#[4]
= #{#{&}#[3]}#
= #{#{#{&}#[2]}#}#
= #{#{#{#}#}#}#
etc.
(#{&}#){&}#[4]
= (#{&}#){(#{&}#){&}#[3]}#
= (#{&}#){(#{&}#){(#{&}#){&}#[2]}#}#
= (#{&}#){(#{&}#){(#{&}#){#{&}#}#}#}#
etc.
So:
E100(#{&}#){&}#1 = E100#{&}#100
E100(#{&}#){&}#2 = E100(#{&}#){#{&}#}#100
E100(#{&}#){&}#3 = E100(#{&}#){(#{&}#){#{&}#}#}#100
E100(#{&}#){&}#4 = E100(#{&}#){(#{&}#){(#{&}#){#{&}#}#}#}#100
E100(#{&}#){&}#5 = E100(#{&}#){(#{&}#){(#{&}#){(#{&}#){#{&}#}#}#}#}#100
etc.
But not:
E100(#{&}#){&}#2 = E100(#{&}#){#{&}#}(#{&}#)100
... due to the ordinal level complexity making it unfamiliar with the Veblen hierarchy in comparison.
On the other hand:
E100(#{&}#){#{&}#}#1 = E100(#{&}#){#}#100
E100(#{&}#){#{&}#}#2 = E100(#{&}#){#{#}#}#100
E100(#{&}#){#{&}#}#3 = E100(#{&}#){#{#{#}#}#}#100
E100(#{&}#){#{&}#}#4 = E100(#{&}#){#{#{#{#}#}#}#}#100
etc.
With the fundamental sequence on caret-tops of %(n)>#:
E100#{&}#>#1 = E100#{&}#100
E100#{&}#>#2 = E100(#{&}#){&}#100
E100#{&}#>#3 = E100((#{&}#){&}#){&}#100
E100#{&}#>#4 = E100(((#{&}#){&}#){&}#){&}#100
etc.
Prepare for the new diagonalization of ampersands...
C. When b= k+& in {}:
C.1. %(a){k+&}#[0] = #
C.2. %(a){k+&}#[1] = %(a){k}#
C.3. %(a){k+&}#[n+1] = %(a){k+(a){k+&}#}#[n] for n > 1
For example:
#{&+&}#[3]
= #{&+#{&+&}#[2]}#
= #{&+#{&+#{&+&}#[1]}#}#
= #{&+#{&+#{&}#}#}#}#
Making the sequence:
E100#{&+&}#1 = E100#{&}#100
E100#{&+&}#2 = E100#{&+#{&}#}#100
E100#{&+&}#3 = E100#{&+#{&+#{&}#}#}#100
E100#{&+&}#4 = E100#{&+#{&+#{&+#{&}#}#}#}#100
etc.
D. When b= k*& in {}:
D.1. %(a){k*&}#[0] = #
D.2. %(a){k*&}#[1] = %(a){k}#
D.3. %(a){k*&}#[n+1] = %(a){k*(a){k*&}#}#[n] for n > 1
For example:
#{&&}#[3]
= #{&#{&&}#[2]}#
= #{&#{&#{&&}#[1]}#}#
= #{&#{&#{&}#}#}#}#
Making the sequence:
E100#{&&}#1 = E100#{&}#100
E100#{&&}#2 = E100#{&#{&}#}#100
E100#{&&}#3 = E100#{&#{&#{&}#}#}#100
E100#{&&}#4 = E100#{&#{&#{&#{&}#}#}#}#100
etc.
E. When b= k^& in {}:
E.1. %(a){k^&}#[0] = #
E.2. %(a){k^&}#[1] = %(a){k}#
E.3. %(a){k^&}#[n+1] = %(a){k^(a){k^&}#}#[n] for n > 1
For example:
#{&^&}#[3]
= #{&^#{&^&}#[2]}#
= #{&^#{&^#{&^&}#[1]}#}#
= #{&^#{&^#{&}#}#}#}#
Making the sequence:
E100#{&^&}#1 = E100#{&}#100
E100#{&^&}#2 = E100#{&^#{&}#}#100
E100#{&^&}#3 = E100#{&^#{&^#{&}#}#}#100
E100#{&^&}#4 = E100#{&^#{&^#{&^#{&}#}#}#}#100
etc.
And finally, the tetrational &, which diagonalizes over the limit of the Collapsing-E notation as follows:
&^^#[1] = &
&^^#[2] = &^&
&^^#[3] = &^&^&
etc.
This plays the similar role to that of the Cascading-E notation's limit delimiter of:
#^^#[1] = #
#^^#[2] = #^#
#^^#[3] = #^#^#
etc.
And that's the limit of the notation is expected to reach the Bachmann-Howard ordinal level in the FGH!
With all the previous formal rules defined by Sbiis Saibian over several years ago remained unchanged!
With the complex example:
E100(#{&^&}#){&&}#3
= E100(#{&^&}#){&&}#[3]100
= E100(#{&^&}#){&(#{&^&}#){&&}#[2]}#100
= E100(#{&^&}#){&(#{&^&}#){&(#{&^&}#){&&}#[1]}#}#100
= E100(#{&^&}#){&(#{&^&}#){&(#{&^&}#){&}#}#}#100
And:
E3#{&^&^&^&#}#3
= E3#{&^&^&^&#}#[3]3
= E3#{&^&^&^(&3)}#3
= E3#{&^&^&^(&+&+&)}#3
= E3#{&^&^(&^&*&^&*&^&)}#3
= E3#{&^&^(&^&*&^&*&^&)}#[3]3
= E3#{&^&^(&^&*&^&*&^#{&^&^(&^&*&^&*&^&)}#[2])}#3
= E3#{&^&^(&^&*&^&*&^#{&^&^(&^&*&^&*&^#{&^&^(&^&*&^&*&^&)}#[1])}#)}#3
= E3#{&^&^(&^&*&^&*&^#{&^&^(&^&*&^&*&^#{&^&^(&^&*&^&*&)}#)}#)}#3
With the rogue-type one:
E100#{#+&}#3
= E100#{#+&}#[3]100
= E100#{#+#{#+&}#[2]}100
= E100#{#+#{#+#{#+&}#[1]}#}100
= E100#{#+#{#+#{#}#}#}100
Introducing the ordinal collapsing function
With the unseen ordinal collapsing functions, in which the first uncountable cardinal in the argument obey in which θ(Ω) is equal to the Ackermann ordinal (φ(1,0,0,0)).
Here we introduce the ordinal collapsing function using "theta" θ function.
A. θ(a, b, c) = φ(a, b, c) using extended Veblen function, for all arguments a < Ω
B. For ordinals a greater or equal to Ω, then θ(a, b, c) is intended to run through fundamental sequences of the Madore's function.
And here, we compare the three OCFs with the extended Buchholz's function, extended Veblen function, and my θ function for each Collapsing-E delimiters.
#{&}# => θ(Ω,0,0) = ψ0(Ω^Ω^2) = φ(1,0,0,0)
(#{&}#)^^# => θ(0,1,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω) = ε(φ(1,0,0,0)+1)
(#{&}#)^^#># => θ(0,1,θ(Ω,0,0)+ω) = ψ0(Ω^Ω^2+Ωω) = ε(φ(1,0,0,0)+ω)
(#{&}#)^^#>#{&}# => θ(0,1,θ(Ω,0,0)2) = ψ0(Ω^Ω^2+Ωψ0(Ω^Ω^2)) = ε(φ(1,0,0,0)2)
(#{&}#)^^#>(#{&}#)^^# => θ(0,1,θ(0,1,θ(Ω,0,0)+1)) = ψ0(Ω^Ω^2+Ωψ0(Ω^Ω^2+Ω)) = ε(ε(φ(1,0,0,0)+1))
(#{&}#)^^## => θ(0,2,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^2) = ζ(φ(1,0,0,0)+1)
(#{&}#)^^### => θ(0,3,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^3) = η(φ(1,0,0,0)+1)
(#{&}#)^^#### => θ(0,4,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^4) = φ(4,φ(1,0,0,0)+1)
(#{&}#)^^#^# => θ(0,ω,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^ω) = φ(ω,φ(1,0,0,0)+1)
(#{&}#)^^(#{&}#) => θ(0,θ(Ω,0,0),1) = ψ0(Ω^Ω^2+Ω^ψ0(Ω^Ω^2)) = φ(φ(1,0,0,0),1)
(#{&}#)^^^# => θ(1,0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ω) = Γ(φ(1,0,0,0)+1)
(#{&}#)^^^## => θ(1,1,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^(Ω+1)) = φ(1,1,φ(1,0,0,0)+1)
(#{&}#)^^^#^# => θ(1,ω,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^(Ω+ω)) = φ(1,ω,φ(1,0,0,0)+1)
(#{&}#)^^^(#{&}#) => θ(1,θ(Ω,0,0),1) = ψ0(Ω^Ω^2+Ω^(Ω+ψ0(Ω^Ω^2))) = φ(1,φ(1,0,0,0),1)
(#{&}#)^^^^# => θ(2,0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ω2) = φ(2,0,φ(1,0,0,0)+1)
(#{&}#)^^^^^# => θ(3,0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ω3) = φ(3,0,φ(1,0,0,0)+1)
(#{&}#)^^^^^^# => θ(4,0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ω4) = φ(4,0,φ(1,0,0,0)+1)
(#{&}#){#}# => θ(ω,0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ωω) = φ(ω,0,φ(1,0,0,0)+1)
(#{&}#){#^#}# => θ(ω^ω,0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ωω^ω) = φ(ω^ω,0,φ(1,0,0,0)+1)
(#{&}#){#^^#}# => θ(θ(0,1,0),0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^ψ0(Ω)) = φ(ε0,0,φ(1,0,0,0)+1)
(#{&}#){#^^##}# => θ(θ(0,2,0),0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^2)) = φ(ζ0,0,φ(1,0,0,0)+1)
(#{&}#){#^^^#}# => θ(θ(1,0,0),0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω)) = φ(Γ0,0,φ(1,0,0,0)+1)
(#{&}#){#{#}#}# => θ(θ(ω,0,0),0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ωω)) = φ(φ(ω,0,0),0,φ(1,0,0,0)+1)
(#{&}#){#{#{#}#}#}# => θ(θ(θ(ω,0,0),0,0),0,θ(Ω,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ωψ0(Ω^Ωω))) = φ(φ(φ(ω,0,0),0,0),0,φ(1,0,0,0)+1)
...
(#{&}#){#{&}#}# => θ(θ(Ω,0,0),0,1) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2)) = φ(φ(1,0,0,0),0,1)
((#{&}#){#{&}#}#){#{&}#}# => θ(θ(Ω,0,0),0,2) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2)2) = φ(φ(1,0,0,0),0,2)
(#{&}#){#{&}#}#># => θ(θ(Ω,0,0),0,ω) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2)ω) = φ(φ(1,0,0,0),0,ω)
(#{&}#){#{&}#}#>#{&}# => θ(θ(Ω,0,0),0,θ(Ω,0,0)) = ψ0(Ω^Ω^2+(Ω^Ωψ0(Ω^Ω^2))ψ0(Ω^Ω^2)) = φ(φ(1,0,0,0),0,φ(1,0,0,0))
(#{&}#){#{&}#}## => θ(θ(Ω,0,0),1,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)+1)) = φ(φ(1,0,0,0),1,0)
(#{&}#){#{&}#}#^# => θ(θ(Ω,0,0),ω,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)+ω)) = φ(φ(1,0,0,0),ω,0)
(#{&}#){#{&}#}(#{&}#) => θ(θ(Ω,0,0),θ(Ω,0,0),0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)+ψ0(Ω^Ω^2))) = φ(φ(1,0,0,0),φ(1,0,0,0),0)
(#{&}#){#{&}#+1}# => θ(θ(Ω,0,0)+1,0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)+Ω)) = φ(φ(1,0,0,0)+1,0,0)
(#{&}#){#{&}#+2}# => θ(θ(Ω,0,0)+2,0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)+Ω2)) = φ(φ(1,0,0,0)+2,0,0)
(#{&}#){#{&}#+#}# => θ(θ(Ω,0,0)+ω,0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)+Ωω)) = φ(φ(1,0,0,0)+ω,0,0)
(#{&}#){#{&}#+#{&}#}# => θ(θ(Ω,0,0)2,0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)2)) = φ(φ(1,0,0,0)2,0,0)
(#{&}#){#{&}#*#}# => θ(θ(Ω,0,0)ω,0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)ω)) = φ(φ(1,0,0,0)ω,0,0)
(#{&}#){#{&}#*#{&}#}# => θ(θ(Ω,0,0)^2,0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)^2)) = φ(φ(1,0,0,0)^2,0,0)
(#{&}#){(#{&}#)^#}# => θ(θ(Ω,0,0)^ω,0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)^ω)) = φ(φ(1,0,0,0)^ω,0,0)
(#{&}#){(#{&}#)^(#{&}#)}# => θ(θ(Ω,0,0)^θ(Ω,0,0),0,0) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2)^ψ0(Ω^Ω^2))) = φ(φ(1,0,0,0)^φ(1,0,0,0),0,0)
(#{&}#){(#{&}#)^^#}# => θ(θ(0,1,θ(Ω,0,0)+1),0,0) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω)) = φ(ε(φ(1,0,0,0)+1),0,0)
(#{&}#){(#{&}#)^^##}# => θ(θ(0,2,θ(Ω,0,0)+1),0,0) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω^2)) = φ(ζ(φ(1,0,0,0)+1),0,0)
(#{&}#){(#{&}#)^^^#}# => θ(θ(1,0,θ(Ω,0,0)+1),0,0) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω^Ω)) = φ(Γ(φ(1,0,0,0)+1),0,0)
(#{&}#){(#{&}#)^^^^#}# => θ(θ(2,0,θ(Ω,0,0)+1),0,0) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω^Ω2)) = φ(φ(2,0,φ(1,0,0,0)+1),0,0)
(#{&}#){(#{&}#){#}#}# => θ(θ(ω,0,θ(Ω,0,0)+1),0,0) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω^Ωω)) = φ(φ(ω,0,φ(1,0,0,0)+1),0,0)
(#{&}#){(#{&}#){#{&}#}#}# => θ(θ(θ(Ω,0,0),0,1),0,0) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2))) = φ(φ(φ(1,0,0,0),0,1),0,0)
(#{&}#){(#{&}#){(#{&}#){#{&}#}#}#}# => θ(θ(θ(θ(Ω,0,0),0,1),0,0),0,0) = ψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2+Ω^Ωψ0(Ω^Ω^2)))) = φ(φ(φ(φ(1,0,0,0),0,1),0,0),0,0)
...
(#{&}#){&}# => θ(Ω,0,1) = ψ0((Ω^Ω^2)2) = φ(1,0,0,1)
((#{&}#){&}#){#{&}#}# => θ(θ(Ω,0,0),0,θ(Ω,0,1)+1) = ψ0((Ω^Ω^2)2+Ω^Ωψ0(Ω^Ω^2))) = φ(φ(1,0,0,0),0,φ(1,0,0,1)+1)
((#{&}#){&}#){(#{&}#){&}#}# => θ(θ(Ω,0,1),0,1) = ψ0((Ω^Ω^2)2+Ω^Ωψ0((Ω^Ω^2)2))) = φ(φ(1,0,0,1),0,1)
((#{&}#){&}#){&}# => θ(Ω,0,2) = ψ0((Ω^Ω^2)3) = φ(1,0,0,2)
(((#{&}#){&}#){&}#){&}# => θ(Ω,0,3) = ψ0((Ω^Ω^2)4) = φ(1,0,0,3)
#{&}#># => θ(Ω,0,ω) = ψ0((Ω^Ω^2)ω) = φ(1,0,0,ω)
#{&}#>#{&}# => θ(Ω,0,θ(Ω,0,0)) = ψ0((Ω^Ω^2)ψ0(Ω^Ω^2)) = φ(1,0,0,φ(1,0,0,0))
...
#{&}## => θ(Ω,1,0) = ψ0(Ω^(Ω^2+1)) = φ(1,0,1,0)
#{&}### => θ(Ω,2,0) = ψ0(Ω^(Ω^2+2)) = φ(1,0,2,0)
#{&}#### => θ(Ω,3,0) = ψ0(Ω^(Ω^2+3)) = φ(1,0,3,0)
#{&}#^# => θ(Ω,ω,0) = ψ0(Ω^(Ω^2+ω)) = φ(1,0,ω,0)
#{&}#{&}# => θ(Ω,θ(Ω,0,0),0) = ψ0(Ω^(Ω^2+ψ0(Ω^Ω^2))) = φ(1,0,φ(1,0,0,0),0)
Moving on...
#{&+1}# => θ(Ω+1,0,0) = ψ0(Ω^(Ω^2+Ω)) = φ(1,1,0,0)
#{&+1}## => θ(Ω+1,1,0) = ψ0(Ω^(Ω^2+Ω+1)) = φ(1,1,1,0)
#{&+2}# => θ(Ω+2,0,0) = ψ0(Ω^(Ω^2+Ω2)) = φ(1,2,0,0)
#{&+3}# => θ(Ω+3,0,0) = ψ0(Ω^(Ω^2+Ω3)) = φ(1,3,0,0)
#{&+#}# => θ(Ω+ω,0,0) = ψ0(Ω^(Ω^2+Ωω)) = φ(1,ω,0,0)
#{&+#^^#}# => θ(Ω+θ(0,1,0),0,0) = ψ0(Ω^(Ω^2+Ωψ0(Ω))) = φ(1,ε0,0,0)
#{&+#{#}#}# => θ(Ω+θ(ω,0,0),0,0) = ψ0(Ω^(Ω^2+Ωψ0(Ω^Ωω))) = φ(1,φ(ω,0,0),0,0)
#{&+#{&}#}# => θ(Ω+θ(Ω,0,0),0,0) = ψ0(Ω^(Ω^2+Ωψ0(Ω^Ω^2))) = φ(1,φ(1,0,0,0),0,0)
#{&+#{&+1}#}# => θ(Ω+θ(Ω+1,0,0),0,0) = ψ0(Ω^(Ω^2+Ωψ0(Ω^(Ω^2+1)))) = φ(1,φ(1,1,0,0),0,0)
#{&+#{&+#}#}# => θ(Ω+θ(Ω+ω,0,0),0,0) = ψ0(Ω^(Ω^2+Ωψ0(Ω^(Ω^2+Ωω)))) = φ(1,φ(1,ω,0,0),0,0)
#{&+#{&+#{&}#}#}# => θ(Ω+θ(Ω+θ(Ω,0,0),0,0),0,0) = ψ0(Ω^(Ω^2+Ωψ0(Ω^(Ω^2+Ωψ0(Ω^Ω^2))))) = φ(1,φ(1,φ(1,0,0,0),0,0),0,0)
...
#{&+&}# => θ(Ω2,0,0) = ψ0(Ω^((Ω^2)2)) = φ(2,0,0,0)
#{&+&+1}# => θ(Ω2+1,0,0) = ψ0(Ω^((Ω^2)2+Ω)) = φ(2,1,0,0)
#{&+&+#}# => θ(Ω2+ω,0,0) = ψ0(Ω^((Ω^2)2+Ωω)) = φ(2,ω,0,0)
#{&+&+#{&+&}#}# => θ(Ω2+θ(Ω2,0,0),0,0) = ψ0(Ω^((Ω^2)2+Ωψ0(Ω^((Ω^2)2)))) = φ(2,φ(2,0,0,0),0,0)
#{&+&+&}# => θ(Ω3,0,0) = ψ0(Ω^((Ω^2)3)) = φ(3,0,0,0)
#{&+&+&+&}# => θ(Ω4,0,0) = ψ0(Ω^((Ω^2)4)) = φ(4,0,0,0)
...
#{&#}# => θ(Ωω,0,0) = ψ0(Ω^((Ω^2)ω)) = φ(ω,0,0,0)
#{&#{&}#}# => θ(Ωθ(Ω,0,0),0,0) = ψ0(Ω^((Ω^2)ψ0(Ω^Ω^2))) = φ(φ(1,0,0,0),0,0,0)
#{&#{&#{&}#}#}# => θ(Ωθ(Ωθ(Ω,0,0),0,0),0,0) = ψ0(Ω^((Ω^2)ψ0(Ω^((Ω^2)ψ0(Ω^Ω^2))))) = φ(φ(φ(1,0,0,0),0,0,0),0,0,0)
...
#{&&}# => θ(Ω^2,0,0) = ψ0(Ω^Ω^3) = φ(1,0,0,0,0)
#{&&+1}# => θ(Ω^2+1,0,0) = ψ0(Ω^(Ω^3+Ω)) = φ(1,0,1,0,0)
#{&&+&}# => θ(Ω^2+Ω,0,0) = ψ0(Ω^(Ω^3+Ω^2)) = φ(1,1,0,0,0)
#{&&+&&}# => θ((Ω^2)2,0,0) = ψ0(Ω^((Ω^3)2)) = φ(2,0,0,0,0)
#{&&#}# => θ((Ω^2)ω,0,0) = ψ0(Ω^((Ω^3)ω)) = φ(ω,0,0,0,0)
#{&&#{&&}#}# => θ((Ω^2)θ(Ω^2,0,0),0,0) = ψ0(Ω^((Ω^3)ψ0(Ω^Ω^3))) = φ(φ(1,0,0,0,0),0,0,0,0)
...
#{&&&}# => θ(Ω^3,0,0) = ψ0(Ω^Ω^4) = φ(1,0,0,0,0,0)
#{&&&+&&&}# => θ((Ω^3)2,0,0) = ψ0(Ω^((Ω^4)2)) = φ(2,0,0,0,0,0)
#{&&&#}# => θ((Ω^3)ω,0,0) = ψ0(Ω^((Ω^4)ω)) = φ(ω,0,0,0,0,0)
#{&&&&}# => θ(Ω^4,0,0) = ψ0(Ω^Ω^5) = φ(1,0,0,0,0,0,0)
#{&&&&&}# => θ(Ω^5,0,0) = ψ0(Ω^Ω^6) = φ(1,0,0,0,0,0,0,0)
...
And here's the limit of the finitary Veblen function! The limit is #{&^#}#.
Let's continue further beyond the small Veblen ordinal!
Here, I am using Rgetar's @ function for the extended transfinitary Veblen function, using curly brackets {...} in place of superscripts above @.
#{&^#}# => θ(Ω^ω,0,0) = ψ0(Ω^Ω^ω) = φ(1 @{ω}) = SVO
*The climbing method catches up at this ordinal
(#{&^#}#){&^#}# => θ(Ω^ω,0,1) = ψ0((Ω^Ω^ω)2) = φ(1 @{ω}, 1)
#{&^#}#># => θ(Ω^ω,0,ω) = ψ0((Ω^Ω^ω)ω) = φ(1 @{ω}, ω)
#{&^#}#>#{&^#}# => θ(Ω^ω,0,θ(Ω^ω,0,0)) = ψ0((Ω^Ω^ω)ψ0(Ω^Ω^ω)) = φ(1 @{ω}, φ(1 @{ω}))
#{&^#}## => θ(Ω^ω,1,0) = ψ0(Ω^(Ω^ω+1)) = φ(1 @{ω}, 1,0)
#{&^#}#^# => θ(Ω^ω,ω,0) = ψ0(Ω^(Ω^ω+ω)) = φ(1 @{ω}, ω,0)
#{&^#}#{&^#}# => θ(Ω^ω,θ(Ω^ω,0,0),0) = ψ0(Ω^(Ω^ω+ψ0(Ω^Ω^ω))) = φ(1 @{ω}, φ(1 @{ω}),0)
#{&^#+1}# => θ(Ω^ω+1,0,0) = ψ0(Ω^(Ω^ω+Ω)) = φ(1 @{ω}, 1,0,0)
#{&^#+#}# => θ(Ω^ω+ω,0,0) = ψ0(Ω^(Ω^ω+Ωω)) = φ(1 @{ω}, ω,0,0)
#{&^#+#{&^#}#}# => θ(Ω^ω+θ(Ω^ω,0,0),0,0) = ψ0(Ω^(Ω^ω+Ωψ0(Ω^Ω^ω))) = φ(1 @{ω}, φ(1 @{ω}),0,0)
#{&^#+&}# => θ(Ω^ω+Ω,0,0) = ψ0(Ω^(Ω^ω+Ω^2)) = φ(1 @{ω}, 1,0,0,0)
#{&^#+&&}# => θ(Ω^ω+Ω^2,0,0) = ψ0(Ω^(Ω^ω+Ω^3)) = φ(1 @{ω}, 1,0,0,0,0)
#{&^#+&^#}# => θ((Ω^ω)2,0,0) = ψ0(Ω^((Ω^ω)2)) = φ(2 @{ω})
#{&^#+&^#+&^#}# => θ((Ω^ω)3,0,0) = ψ0(Ω^((Ω^ω)3)) = φ(3 @{ω})
#{&^#*#}# => θ((Ω^ω)ω,0,0) = ψ0(Ω^((Ω^ω)ω)) = φ(ω @{ω})
#{&^#*#{&^#}#}# => θ((Ω^ω)θ(Ω^ω,0,0),0,0) = ψ0(Ω^((Ω^ω)ψ0(Ω^Ω^ω))) = φ(φ(1 @{ω}) @{ω})
...
#{&^#*&}# => θ(Ω^(ω+1),0,0) = ψ0(Ω^Ω^(ω+1)) = φ(1,0 @{ω})
#{&^#*&&}# => θ(Ω^(ω+2),0,0) = ψ0(Ω^Ω^(ω+2)) = φ(1,0,0 @{ω})
#{&^#*&&&}# => θ(Ω^(ω+3),0,0) = ψ0(Ω^Ω^(ω+3)) = φ(1,0,0,0 @{ω})
#{&^#*&^#}# => θ(Ω^ω2,0,0) = ψ0(Ω^Ω^ω2) = φ(1 @{ω2})
#{&^#*&^#*&^#}# => θ(Ω^ω3,0,0) = ψ0(Ω^Ω^ω3) = φ(1 @{ω3})
...
#{&^##}# => θ(Ω^ω^2,0,0) = ψ0(Ω^Ω^ω^2) = φ(1 @{ω^2})
#{&^###}# => θ(Ω^ω^3,0,0) = ψ0(Ω^Ω^ω^3) = φ(1 @{ω^3})
#{&^####}# => θ(Ω^ω^4,0,0) = ψ0(Ω^Ω^ω^4) = φ(1 @{ω^4})
...
#{&^#^#}# => θ(Ω^ω^ω,0,0) = ψ0(Ω^Ω^ω^ω) = φ(1 @{ω^ω})
#{&^#^##}# => θ(Ω^ω^ω^2,0,0) = ψ0(Ω^Ω^ω^ω^2) = φ(1 @{ω^ω^2})
#{&^#^#^#}# => θ(Ω^ω^ω^ω,0,0) = ψ0(Ω^Ω^ω^ω^ω) = φ(1 @{ω^ω^ω})
#{&^#^#^#^#}# => θ(Ω^ω^ω^ω^ω,0,0) = ψ0(Ω^Ω^ω^ω^ω^ω) = φ(1 @{ω^ω^ω^ω})
...
#{&^#^^#}# => θ(Ω^θ(0,1,0),0,0) = ψ0(Ω^Ω^ψ0(Ω)) = φ(1 @{ε0})
#{&^#^^##}# => θ(Ω^θ(0,2,0),0,0) = ψ0(Ω^Ω^ψ0(Ω^2)) = φ(1 @{ζ0})
#{&^#^^^#}# => θ(Ω^θ(1,0,0),0,0) = ψ0(Ω^Ω^ψ0(Ω^Ω)) = φ(1 @{Γ0})
#{&^#{&}#}# => θ(Ω^θ(Ω,0,0),0,0) = ψ0(Ω^Ω^ψ0(Ω^Ω^2)) = φ(1 @{φ(1,0,0,0)})
#{&^#{&&}#}# => θ(Ω^θ(Ω^2,0,0),0,0) = ψ0(Ω^Ω^ψ0(Ω^Ω^3)) = φ(1 @{φ(1,0,0,0,0)})
#{&^#{&^#}#}# => θ(Ω^θ(Ω^ω,0,0),0,0) = ψ0(Ω^Ω^ψ0(Ω^Ω^ω)) = φ(1 @{φ(1 @{ω})})
#{&^#{&^#{&}#}#}# => θ(Ω^θ(Ω^θ(Ω,0,0),0,0),0,0) = ψ0(Ω^Ω^ψ0(Ω^Ω^ψ0(Ω^Ω^2))) = φ(1 @{φ(1 @{φ(1,0,0,0)})})
...
Now we reach the large Veblen ordinal!
#{&^&}# => θ(Ω^Ω,0,0) = ψ0(Ω^Ω^Ω) = φ(1 @{1,0})
#{&^&}## => θ(Ω^Ω,1,0) = ψ0(Ω^(Ω^Ω+1)) = φ(1 @{1,0}, 1,0)
#{&^&+1}# => θ(Ω^Ω+1,0,0) = ψ0(Ω^(Ω^Ω+Ω)) = φ(1 @{1,0}, 1,0,0)
#{&^&+&}# => θ(Ω^Ω+Ω,0,0) = ψ0(Ω^(Ω^Ω+Ω^2)) = φ(1 @{1,0}, 1,0,0,0)
#{&^&+&^#}# => θ(Ω^Ω+Ω^ω,0,0) = ψ0(Ω^(Ω^Ω+Ω^ω)) = φ(1 @{1,0}, 1 @{ω} 0)
#{&^&+&^#{&^&}#}# => θ(Ω^Ω+Ω^θ(Ω^Ω,0,0),0,0) = ψ0(Ω^(Ω^Ω+Ω^ψ0(Ω^Ω^Ω))) = φ(1 @{1,0}, 1 @{φ(1 @{1,0})} 0)
#{&^&+&^&}# => θ((Ω^Ω)2,0,0) = ψ0(Ω^((Ω^Ω)2)) = φ(2 @{1,0})
#{&^&*#}# => θ((Ω^Ω)ω,0,0) = ψ0(Ω^((Ω^Ω)ω)) = φ(ω @{1,0})
#{&^&*#{&^&}#}# => θ((Ω^Ω)θ(Ω^Ω,0,0),0,0) = ψ0(Ω^((Ω^Ω)ψ0(Ω^Ω^Ω))) = φ(φ(1 @{1,0}) @{1,0})
#{&^&*&}# => θ(Ω^(Ω+1),0,0) = ψ0(Ω^Ω^(Ω+1)) = φ(1,0 @{1,0})
#{&^&*&&}# => θ(Ω^(Ω+2),0,0) = ψ0(Ω^Ω^(Ω+2)) = φ(1,0,0 @{1,0})
#{&^&*&^#}# => θ(Ω^(Ω+ω),0,0) = ψ0(Ω^Ω^(Ω+ω)) = φ(1 @{1,ω})
#{&^&*&^#{&^&}#}# => θ(Ω^(Ω+θ(Ω^Ω,0,0)),0,0) = ψ0(Ω^Ω^(Ω+ψ0(Ω^Ω^Ω))) = φ(1 @{1,φ(1 @{1,0})})
#{&^&*&^&}# => θ(Ω^Ω2,0,0) = ψ0(Ω^Ω^Ω2) = φ(1 @{2,0})
#{&^&#}# => θ(Ω^Ωω,0,0) = ψ0(Ω^Ω^Ωω) = φ(1 @{ω,0})
#{&^&#{&^&}#}# => θ(Ω^Ωθ(Ω^Ω,0,0),0,0) = ψ0(Ω^Ω^Ωψ0(Ω^Ω^Ω)) = φ(1 @{φ(1 @{1,0}),0})
...
#{&^&&}# => θ(Ω^Ω^2,0,0) = ψ0(Ω^Ω^Ω^2) = φ(1 @{1,0,0})
#{&^&&*&}# => θ(Ω^(Ω^2+1),0,0) = ψ0(Ω^Ω^(Ω^2+1)) = φ(1,0 @{1,0,0})
#{&^&&*&^&}# => θ(Ω^(Ω^2+Ω),0,0) = ψ0(Ω^Ω^(Ω^2+Ω)) = φ(1 @{1,1,0})
#{&^&&*&^&#}# => θ(Ω^(Ω^2+Ωω),0,0) = ψ0(Ω^Ω^(Ω^2+Ωω)) = φ(1 @{1,ω,0})
#{&^&&*&^&&}# => θ(Ω^((Ω^2)2),0,0) = ψ0(Ω^Ω^((Ω^2)2)) = φ(1 @{2,0,0})
#{&^&&#}# => θ(Ω^((Ω^2)ω),0,0) = ψ0(Ω^Ω^((Ω^2)ω)) = φ(1 @{ω,0,0})
#{&^&&#{&^&&}#}# => θ(Ω^((Ω^2)θ(Ω^Ω^2,0,0)),0,0) = ψ0(Ω^Ω^((Ω^2)ψ0(Ω^Ω^Ω^2))) = φ(1 @{φ(1 @{1,0,0}),0,0})
...
#{&^&&&}# => θ(Ω^Ω^3,0,0) = ψ0(Ω^Ω^Ω^3) = φ(1 @{1,0,0,0})
#{&^&&&&}# => θ(Ω^Ω^4,0,0) = ψ0(Ω^Ω^Ω^4) = φ(1 @{1,0,0,0,0})
#{&^&&&&&}# => θ(Ω^Ω^5,0,0) = ψ0(Ω^Ω^Ω^5) = φ(1 @{1,0,0,0,0,0})
...
#{&^&^#}# => θ(Ω^Ω^ω,0,0) = ψ0(Ω^Ω^Ω^ω) = φ(1 @{1 @{ω}})
#{&^&^#{&^&}#}# => θ(Ω^Ω^θ(Ω^Ω,0,0),0,0) = ψ0(Ω^Ω^Ω^ψ0(Ω^Ω^Ω)) = φ(1 @{1 @{φ(1 @{1,0})}})
#{&^&^#{&^&^#{&^&}#}#}# => θ(Ω^Ω^θ(Ω^Ω^θ(Ω^Ω,0,0),0,0),0,0) = ψ0(Ω^Ω^Ω^ψ0(Ω^Ω^Ω^ψ0(Ω^Ω^Ω))) = φ(1 @{1 @{φ(1 @{1 @{1, 0}})}})
...
#{&^&^&}# => θ(Ω^Ω^Ω,0,0) = ψ0(Ω^Ω^Ω^Ω) = φ(1 @{1 @{1,0}})
#{&^&^&&}# => θ(Ω^Ω^Ω^2,0,0) = ψ0(Ω^Ω^Ω^Ω^2) = φ(1 @{1 @{1,0,0}})
#{&^&^&&&}# => θ(Ω^Ω^Ω^3,0,0) = ψ0(Ω^Ω^Ω^Ω^3) = φ(1 @{1 @{1,0,0,0}})
#{&^&^&^#}# => θ(Ω^Ω^Ω^ω,0,0) = ψ0(Ω^Ω^Ω^Ω^ω) = φ(1 @{1 @{1 @{ω}}})
#{&^&^&^#{&^&^&^&}#}# => θ(Ω^Ω^Ω^θ(Ω^Ω^Ω,0,0),0,0) = ψ0(Ω^Ω^Ω^Ω^ψ0(Ω^Ω^Ω^Ω)) = φ(1 @{1 @{1 @{φ(1 @{1 @{1,0}})}}})
...
#{&^&^&^&}# => θ(Ω^Ω^Ω^Ω,0,0) = ψ0(Ω^Ω^Ω^Ω^Ω) = φ(1 @{1 @{1 @{1,0}}})
#{&^&^&^&^#}# => θ(Ω^Ω^Ω^Ω^ω,0,0) = ψ0(Ω^Ω^Ω^Ω^Ω^ω) = φ(1 @{1 @{1 @{1 @{ω}}}})
#{&^&^&^&^&}# => θ(Ω^Ω^Ω^Ω^Ω,0,0) = ψ0(Ω^Ω^Ω^Ω^Ω^Ω) = φ(1 @{1 @{1 @{1 @{1,0}}}})
#{&^&^&^&^&^&}# => θ(Ω^Ω^Ω^Ω^Ω^Ω,0,0) = ψ0(Ω^Ω^Ω^Ω^Ω^Ω^Ω) = φ(1 @{1 @{1 @{1 @{1 @{1,0}}}}})
#{&^&^&^&^&^&^&}# => θ(Ω^Ω^Ω^Ω^Ω^Ω^Ω,0,0) = ψ0(Ω^Ω^Ω^Ω^Ω^Ω^Ω^Ω) = φ(1 @{1 @{1 @{1 @{1 @{1 @{1,0}}}}}})
#{&^&^&^&^&^&^&^&}# => θ(Ω^Ω^Ω^Ω^Ω^Ω^Ω^Ω,0,0) = ψ0(Ω^Ω^Ω^Ω^Ω^Ω^Ω^Ω^Ω) = φ(1 @{1 @{1 @{1 @{1 @{1 @{1 @{1,0}}}}}}})
...
And finally...
The limit of the level is:
#{&^^#}# = #{&_2}# = θ(ε(Ω+1),0,0) = ψ0(ε(Ω+1)) = ψ0(Ω_2) = Bachmann-Howard ordinal (BHO)
Another properties of the theta function...
θ(Ω^n,0,0) = ψ0(Ω^Ω^(1+n)) using (extended) Buchholz's function
θ(Ω^n,0,0) = φ(1 @{2+n}) using extended transfinitary Veblen function, for n < Ω, using "@", as φ(1 @{1}) = φ(1,0), φ(1 @{2}) = φ(1,0,0), φ(1 @{3}) = φ(1,0,0,0), and so on.