Hang on, how I assign the limit of the Collapsing-E notation to as #{&^^#}# = #{&_2}#? The reasoning is very simple, as I explained above.
On the other hand, if we continue further with (&^^#)^^#, you'd find out that the fundamental sequence would be supposed to satisfy weird fundamental sequences and growth rates of #{&_2}#, which has the fundamental sequence of &_2[n+1] = &{&_2[n]}#, where &_2[1] = &.
Before defining the actual Extended Collapsing-E notation, I am going to propose the rogue-type ideas of my extension. First, we have to define #{&^^#+&^^#}#.
We observed my largest number ever coined called "blasteriagulus" from the limit of the Collapsing-E notation, and is defined as follows:
(1) blasteriagulus = E100#{&_2}#100 = E100#{&^^#}#100
= E100#{&^&^&^&^... ... ... ...^&^&^&^&}#100
w/ 100 &'s
Of course, we could also have:
(2) blasteriagulgong = E100,000#{&_2}#100,000 = E100#{&^^#}#100,000
(3) grand blasteriagulus = E100#{&_2}#100#2 = E100#{&^^#}#100#2
(4) granblasteriagulus / (5) grangol-carta-blasteriagulus = E100#{&_2}#100#100 = E100#{&^^#}#100#100
(6) blasteriagulatri / (7) blasteriagulus-by-deuteron = E100#{&_2}#100#{&_2}#100 = E100#{&^^#}#100#{&^^#}#100
(8) blasteriagulus-by-hyperion = E100#{&_2}#*#100 = E100#{&^^#}#*#100
(9) deutero-blasteriagulus = E100#{&_2}#*#{&_2}100 = E100#{&^^#}#*#{&^^#}#100
(10) blasteriagulfact = E100(#{&_2}#)^#100 = E100(#{&^^#}#)^#100
(11) dutetrated-blasteriagulus = E100(#{&_2}#)^(#{&_2}#)100 = E100(#{&^^#}#)^(#{&^^#}#)100
...
(12) terrible blasteriagulus = E100(#{&_2}#)^^#100 = E100(#{&^^#}#)^^#100
(13) terrisquared blasteriagulus = E100(#{&_2}#)^^##100 = E100(#{&^^#}#)^^##100
(14) horrible blasteriagulus = E100(#{&_2}#)^^^#100 = E100(#{&^^#}#)^^^#100
(15) horrendous blasteriagulus = E100(#{&_2}#)^^^^#100 = E100(#{&^^#}#)^^^^#100
(16) grievous blasteriagulus = E100(#{&_2}#){#}#100 = E100(#{&^^#}#){#}#100
(17) blasphemorgulnumus blasteriagulus = E100(#{&_2}#){#{&}#}#100 = E100(#{&^^#}#){#{&}#}#100
(18) blasphemous blasteriagulus = E100(#{&_2}#){#{&}#}#100 = E100(#{&^^#}#){#{&}#}#100
(19) blasteriatetrated blasteriagulus = E100(#{&^^#}#){&^^#}#100 = E100(#{&_2}#){&_2}#100?
Hang on... How I stop at E100(#{&^^#}#){&^^#}#100?
Let see that the notation will get weird...
Before defining the extended Collapsing-E notation, I need some fundamental sequence for the limit of (#{&_2}#){&^&^&^...^&^&^&}# with n &'s in the power tower.
Now, what happens if we try this with the extended Collapsing-E notation? Why do we need (&_2)_1? Doing the limit of (#{&}#){#{#{#{...#{#{#}#}#...}#}#}#}# with n #'s from the left-hand side inside {} after (#{&}#) let us to (#{&}#)[#{&}#}#, so does indeed make sense to conjecture the limit of (#{&_2}#){&^&^&^...^&^&^&}# would yield (#{&_2}#){&_2}#, but isn't actually in that case! So we have to define the limit to be (#{&_2}#){(&_2)_1}#! Similarly, the limit of #{&_2+&^&^&^...^&^&^&}# is not #{&_2+&_2}#, but #{&_2+(&_2)_1}#! What we should do?
First, we revise the previous rules of the Collapsing-E notation, which resemble rules of the extended Cascading-E notation.
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 %
I propose to make the limit of the notation's growth rate to be f_{ψ0(Ω_ω)}(n) with respect to the extended Buchholz hierarchy, which is a large countable ordinal that is the proof theoretic ordinal of Π1_1-CA0 subsystem.
We define that & symbol is the shorthand for &_1.
Before introducing the formal definition of the notation, we observe that we can proceed beyond the Bachmann-Howard ordinal (BHO) by introducing the #{&_2}# delimiter. In terms of the Extended Collapsing-E notation, it is defined as:
#{&_2}#[n] = #{&^^#}#[n] = #{&^^n}# = #{&^&^&^&^...^&^&^&^&}# with n &'s
The ordinal levels of the revised separators from #{&_2}# onwards are as follows:
#{&_2}# has ordinal level ψ0(Ω_2) = BHO
(#{&_2}#)^^# has ordinal level ψ0(Ω_2+Ω) = ε(BHO+1)
((#{&_2}#)^^#)^^# has ordinal level ψ0(Ω_2+Ω2) = ε(BHO+2)
(#{&_2}#)^^#># has ordinal level ψ0(Ω_2+Ωω) = ε(BHO+ω)
(#{&_2}#)^^#>#{&_2}# has ordinal level ψ0(Ω_2+Ωψ0(Ω_2)) = ε(BHO2)
(#{&_2}#)^^#>(#{&_2}#)^^# has ordinal level ψ0(Ω_2+Ωψ0(Ω_2+Ω)) = εε(BHO+1)
(#{&_2}#)^^## has ordinal level ψ0(Ω_2+Ω^2) = ζ(BHO+1)
(#{&_2}#)^^### has ordinal level ψ0(Ω_2+Ω^3) = η(BHO+1)
(#{&_2}#)^^#^# has ordinal level ψ0(Ω_2+Ω^ω) = φ(ω, BHO+1)
(#{&_2}#)^^(#{&_2}#) has ordinal level ψ0(Ω_2+Ω^ψ0(Ω_2)) = φ(BHO, 1)
(#{&_2}#)^^^# has ordinal level ψ0(Ω_2+Ω^Ω) = Γ(BHO+1)
(#{&_2}#)^^^^# has ordinal level ψ0(Ω_2+Ω^Ω2) = φ(2, 0, BHO+1)
(#{&_2}#){#}# has ordinal level ψ0(Ω_2+Ω^Ωω) = φ(ω, 0, BHO+1)
(#{&_2}#){#{&_2}#}# has ordinal level ψ0(Ω_2+Ω^Ωψ0(Ω_2)) = φ(BHO, 0, 1)
(#{&_2}#){(#{&_2}#){#{&_2}#}#}# has ordinal level ψ0(Ω_2+Ω^Ωψ0(Ω_2+Ω^Ωψ0(Ω_2))) = φ(φ(BHO, 0, 1), 0, 0)
(#{&_2}#){&}# has ordinal level ψ0(Ω_2+Ω^Ω^2) = φ(1, 0, 0, BHO + 1)
(#{&_2}#){&&}# has ordinal level ψ0(Ω_2+Ω^Ω^3) = φ(1, 0, 0, 0, BHO + 1)
(#{&_2}#){&^#}# has ordinal level ψ0(Ω_2+Ω^Ω^ω) = φ(1 @{ω}, BHO + 1)
(#{&_2}#){&^#{&_2}#}# has ordinal level ψ0(Ω_2+Ω^Ω^ψ0(Ω_2)) = φ(1 @{BHO}, 1)
(#{&_2}#){&^(#{&_2}#){&^#{&_2}#}#}# has ordinal level ψ0(Ω_2+Ω^Ω^ψ0(Ω_2+Ω^Ω^ψ0(Ω_2))) = φ(1 @{φ(1 @{BHO}, 1)}, 0)
(#{&_2}#){&^&}# has ordinal level ψ0(Ω_2+Ω^Ω^Ω) = φ(1 @{1, 0}, BHO+1)
(#{&_2}#){&^&^&}# has ordinal level ψ0(Ω_2+Ω^Ω^Ω^Ω) = φ(1 @{1 @{1, 0}}, BHO+1)
With the (#{&_2}#){(&_2)_1}# delimiter, which has ordinal level ψ0(Ω_2+ψ1(Ω_2)) = ψ0(Ω_2+ε(Ω+1)):
(#{&_2}#){(&_2)_1}#[n] = (#{&_2}#){&^^#}#[n] = (#{&_2}#){&^^n}#
= (#{&_2}#){&^&^&^...^&^&^&}# with n &'s after #{&_2}#
The ordinal levels of some higher delimiters are as follows:
(#{&_2}#){(&_2)_1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1))
((#{&_2}#){(&_2)_1}#)^^# has ordinal level ψ0(Ω_2 + ψ1(Ω_2) + Ω) = ε(ψ0(Ω_2 + ε(Ω + 1)) + 1)
((#{&_2}#){(&_2)_1}#){(&_2)_1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)2) = ψ0(Ω_2 + ε(Ω + 1)2)
(#{&_2}#){(&_2)_1}#># has ordinal level ψ0(Ω_2 + ψ1(Ω_2)ω) = ψ0(Ω_2 + ε(Ω + 1)ω)
(#{&_2}#){(&_2)_1}#>#{&_2}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)ψ0(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)ψ0(Ω_2))
(#{&_2}#){(&_2)_1}#>(#{&_2}#){(&_2)_1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)ψ0(Ω_2 + ψ1(Ω_2))) = ψ0(Ω_2 + ε(Ω + 1)ψ0(Ω_2 + ε(Ω + 1)))
(#{&_2}#){(&_2)_1}## has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω) = ψ0(Ω_2 + ε(Ω + 1)Ω)
(#{&_2}#){(&_2)_1}### has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^2) = ψ0(Ω_2 + ε(Ω + 1)Ω^2)
(#{&_2}#){(&_2)_1}#^# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^ω) = ψ0(Ω_2 + ε(Ω + 1)Ω^ω)
(#{&_2}#){(&_2)_1}(#{&_2}#) has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^ψ0(Ω_2)) = ψ0(Ω_2 + ε(Ω+1)Ω^ψ0(Ω_2))
(#{&_2}#){(&_2)_1+1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω)
(#{&_2}#){(&_2)_1+2}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω2) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω2)
(#{&_2}#){(&_2)_1+#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ωω) = ψ0(Ω_2 + ε(Ω+1)Ω^Ωω)
(#{&_2}#){(&_2)_1+#{&_2}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ωψ0(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ωψ0(Ω_2))
(#{&_2}#){(&_2)_1+(#{&_2}#){(&_2)_1}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ωψ0(Ω_2 + ψ1(Ω_2))) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ωψ0(Ω_2 + ε(Ω + 1)))
(#{&_2}#){(&_2)_1+&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^2) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^2)
(#{&_2}#){(&_2)_1+&&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^3) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^3)
(#{&_2}#){(&_2)_1+&^#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^ω) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^ω)
(#{&_2}#){(&_2)_1+&^#{&_2}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^ψ0(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^ψ0(Ω_2))
(#{&_2}#){(&_2)_1+&^#{&_2}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^ψ0(Ω_2 + ψ1(Ω_2))) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^ψ0(Ω_2))
(#{&_2}#){(&_2)_1+&^(#{&_2}#){&_2}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2)))) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^ψ0(Ω_2 + ε(Ω + 1)))
(#{&_2}#){(&_2)_1+&^&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^Ω) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^Ω)
(#{&_2}#){(&_2)_1+&^&^&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)Ω^Ω^Ω^Ω) = ψ0(Ω_2 + ε(Ω + 1)Ω^Ω^Ω^Ω)
(#{&_2}#){(&_2)_1+(&_2)_1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^2) = ψ0(Ω_2 + ε(Ω + 1)^2)
(#{&_2}#){(&_2)_1+(&_2)_1+(&_2)_1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^3) = ψ0(Ω_2 + ε(Ω + 1)^3)
(#{&_2}#){(&_2)_1*#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ω) = ψ0(Ω_2 + ε(Ω + 1)^ω)
(#{&_2}#){(&_2)_1*#{&_2}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ0(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)^ψ0(Ω_2))
(#{&_2}#){(&_2)_1*(#{&_2}#){(&_2)_1}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ0(Ω_2 + ψ1(Ω_2))) = ψ0(Ω_2 + ε(Ω + 1)^ψ0(Ω_2 + ε(Ω + 1)))
(#{&_2}#){(&_2)_1*&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^Ω) = ψ0(Ω_2 + ε(Ω + 1)^Ω)
(#{&_2}#){(&_2)_1*&&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^Ω^2) = ψ0(Ω_2 + ε(Ω + 1)^Ω^2)
(#{&_2}#){(&_2)_1*&^#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^Ω^ω) = ψ0(Ω_2 + ε(Ω + 1)^Ω^ω)
(#{&_2}#){(&_2)_1*&^&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^Ω^Ω) = ψ0(Ω_2 + ε(Ω + 1)^Ω^Ω)
(#{&_2}#){(&_2)_1*&^&^&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^Ω^Ω^Ω) = ψ0(Ω_2 + ε(Ω + 1)^Ω^Ω^Ω)
(#{&_2}#){(&_2)_1*(&_2)_1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1))
(#{&_2}#){(&_2)_1*(&_2)_1*(&_2)_1}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^2) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^2)
(#{&_2}#){((&_2)_1)^#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ω) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ω)
(#{&_2}#){((&_2)_1)^#{&_2}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ0(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ψ0(Ω_2))
(#{&_2}#){((&_2)_1)^(#{&_2}#){(&_2)_1}#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ0(Ω_2 + ψ1(Ω_2))) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ψ0(Ω_2 + ε(Ω + 1)))
(#{&_2}#){((&_2)_1)^&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^Ω) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^Ω)
(#{&_2}#){((&_2)_1)^&^&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^Ω^Ω) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^Ω^Ω)
(#{&_2}#){((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1))
(#{&_2}#){((&_2)_1)^((&_2)_1)^#}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)^ω) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1)^ω)
(#{&_2}#){((&_2)_1)^((&_2)_1)^&}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)^Ω) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1)^Ω)
(#{&_2}#){((&_2)_1)^((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1))
(#{&_2}#){((&_2)_1)^((&_2)_1)^((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)) = ψ0(Ω_2 + ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1))
Once we observed the pattern of (#{&_2}#){((&_2)_1)^((&_2)_1)^((&_2)_1)^...^((&_2)_1)^((&_2)_1)^((&_2)_1)}# with n ((&_2)_1)'s, this is equal to (#{&_2}#){&_2}#! Yikes!
(#{&_2}#){&_2}# has ordinal level ψ0(Ω_2·2)
((#{&_2}#){&_2}#)^^# has ordinal level ψ0(Ω_2·2 + Ω)
((#{&_2}#){&_2}#){(&_2)_1}# has ordinal level ψ0(Ω_2·2 + ψ1(Ω_2)) = ψ0(Ω_2·2 + ε(Ω + 1))
((#{&_2}#){&_2}#){(&_2)_1+(&_2)_1}# has ordinal level ψ0(Ω_2·2 + ψ1(Ω_2)^2) = ψ0(Ω_2·2 + ε(Ω + 1)^2)
((#{&_2}#){&_2}#){(&_2)_1*(&_2)_1}# has ordinal level ψ0(Ω_2·2 + ψ1(Ω_2)^ψ1(Ω_2)) = ψ0(Ω_2·2 + ε(Ω + 1)^ε(Ω + 1))
((#{&_2}#){&_2}#){((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_2·2 + ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2)) = ψ0(Ω_2·2 + ε(Ω + 1)^ε(Ω + 1)^ε(Ω + 1))
((#{&_2}#){&_2}#){&_2}# has ordinal level ψ0(Ω_2·3)
(((#{&_2}#){&_2}#){&_2}#){&_2}# has ordinal level ψ0(Ω_2·4)
#{&_2}#># has ordinal level ψ0(Ω_2·ω)
#{&_2}#>#{&_2}# has ordinal level ψ0(Ω_2·ψ0(Ω_2))
#{&_2}#>#{&_2}#>#{&_2}# has ordinal level ψ0(Ω_2·ψ0(Ω_2·ψ0(Ω_2)))
#{&_2}## has ordinal level ψ0(Ω_2·Ω)
#{&_2}### has ordinal level ψ0(Ω_2·Ω^2)
#{&_2}#^# has ordinal level ψ0(Ω_2·Ω^ω)
#{&_2}#{&_2}# has ordinal level ψ0(Ω_2·Ω^ψ0(Ω_2))
#{&_2}#{&_2}#{&_2}# has ordinal level ψ0(Ω_2·Ω^ψ0(Ω_2·Ω^ψ0(Ω_2)))
#{&_2+1}# has ordinal level ψ0(Ω_2·Ω^Ω)
#{&_2+2}# has ordinal level ψ0(Ω_2·Ω^Ω2)
#{&_2+#}# has ordinal level ψ0(Ω_2·Ω^Ωω)
#{&_2+#{&_2}#}# has ordinal level ψ0(Ω_2·Ω^Ωψ0(Ω_2))
#{&_2+&}# has ordinal level ψ0(Ω_2·Ω^Ω^2)
#{&_2+&^#}# has ordinal level ψ0(Ω_2·Ω^Ω^ω)
#{&_2+&^&}# has ordinal level ψ0(Ω_2·Ω^Ω^Ω)
#{&_2+&^&^&}# has ordinal level ψ0(Ω_2·Ω^Ω^Ω^Ω)
#{&_2+(&_2)_1}# has ordinal level ψ0(Ω_2·ψ1(Ω_2)) = ψ0(Ω_2·ε(Ω + 1))
#{&_2+(&_2)_1+(&_2)_1}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·2)) = ψ0(Ω_2·ε(Ω + 2))
#{&_2+(&_2)_1*#}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·ω)) = ψ0(Ω_2·ε(Ω + ω))
#{&_2+(&_2)_1*&}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·Ω)) = ψ0(Ω_2·ε(Ω2))
#{&_2+(&_2)_1*(&_2)_1}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·ψ1(Ω_2))) = ψ0(Ω_2·ε(ε(Ω + 1)))
#{&_2+((&_2)_1)^#}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·ψ1(Ω_2·ω))) = ψ0(Ω_2·ε(ε(Ω + ω)))
#{&_2+((&_2)_1)^&}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·ψ1(Ω_2·Ω))) = ψ0(Ω_2·ε(ε(Ω2)))
#{&_2+((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·ψ1(Ω_2·ψ1(Ω_2)))) = ψ0(Ω_2·ε(ε(ε(Ω + 1))))
#{&_2+((&_2)_1)^((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·ψ1(Ω_2·ψ1(Ω_2·ψ1(Ω_2))))) = ψ0(Ω_2·ε(ε(ε(ε(Ω + 1)))))
#{&_2+((&_2)_1)^((&_2)_1)^((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_2·ψ1(Ω_2·ψ1(Ω_2·ψ1(Ω_2·ψ1(Ω_2·ψ1(Ω_2)))))) = ψ0(Ω_2·ε(ε(ε(ε(ε(Ω + 1))))))
... Moving on an another new &_2...
#{&_2+&_2}# has ordinal level ψ0(Ω_2^2)
(#{&_2+&_2}#){&_2}# has ordinal level ψ0(Ω_2^2 + Ω_2)
(#{&_2+&_2}#){&_2+(&_2)_1}# has ordinal level ψ0(Ω_2^2 + ψ1(Ω_2)) = ψ0(Ω_2^2 + ε(Ω + 1))
(#{&_2+&_2}#){&_2+&_2}# has ordinal level ψ0(Ω_2^2·2)
#{&_2+&_2}#># has ordinal level ψ0(Ω_2^2·ω)
#{&_2+&_2}#>#{&_2+&_2}# has ordinal level ψ0(Ω_2^2·ψ0(Ω_2^2·ω))
#{&_2+&_2}## has ordinal level ψ0(Ω_2^2·Ω)
#{&_2+&_2}### has ordinal level ψ0(Ω_2^2·Ω^2)
#{&_2+&_2}#^# has ordinal level ψ0(Ω_2^2·Ω^ω)
#{&_2+&_2}#{&_2+&_2}# has ordinal level ψ0(Ω_2^2·Ω^ψ0(Ω_2^2))
#{&_2+&_2+1}# has ordinal level ψ0(Ω_2^2·Ω^Ω)
#{&_2+&_2+#}# has ordinal level ψ0(Ω_2^2·Ω^Ωω)
#{&_2+&_2+&}# has ordinal level ψ0(Ω_2^2·Ω^Ω^2)
#{&_2+&_2+&^&}# has ordinal level ψ0(Ω_2^2·Ω^Ω^Ω)
#{&_2+&_2+(&_2)_1}# has ordinal level ψ0(Ω_2^2·ψ1(Ω_2)) = ψ0(Ω_2^2·ε(Ω + 1))
#{&_2+&_2+(&_2)_1+(&_2)_1}# has ordinal level ψ0(Ω_2^2·ψ1(Ω_2·2)) = ψ0(Ω_2^2·ε(Ω + 2))
#{&_2+&_2+(&_2)_1*(&_2)_1}# has ordinal level ψ0(Ω_2^2·ψ1(Ω_2·ψ1(Ω_2))) = ψ0(Ω_2^2·ε(ε(Ω + 1)))
#{&_2+&_2+&_2}# has ordinal level ψ0(Ω_2^3)
#{&_2+&_2+&_2+&_2}# has ordinal level ψ0(Ω_2^4)
#{&_2+&_2+&_2+&_2+&_2}# has ordinal level ψ0(Ω_2^5)
#{&_2*#}# has ordinal level ψ0(Ω_2^ω)
#{&_2*#{&_2}#}# has ordinal level ψ0(Ω_2^ψ0(Ω_2^ω))
#{&_2*&}# has ordinal level ψ0(Ω_2^Ω)
#{&_2*(&_2)_1}# has ordinal level ψ0(Ω_2^ψ1(Ω_2)) = ψ0(Ω_2^ε(Ω + 1))
#{&_2*&_2}# has ordinal level ψ0(Ω_2^Ω_2) = ψ0(Ω_2^Ω_2)
#{&_2*&_2*&_2}# has ordinal level ψ0(Ω_2^Ω_2^2) = ψ0(Ω_2^Ω_2^2)
#{&_2*&_2*&_2*&_2}# has ordinal level ψ0(Ω_2^Ω_2^3) = ψ0(Ω_2^Ω_2^3)
#{(&_2)^#}# has ordinal level ψ0(Ω_2^Ω_2^ω) = ψ0(Ω_2^Ω_2^ω)
#{(&_2)^#{&_2}#}# has ordinal level ψ0(Ω_2^Ω_2^ψ0(Ω_2)) = ψ0(Ω_2^Ω_2^ψ0(Ω_2))
#{(&_2)^&}# has ordinal level ψ0(Ω_2^Ω_2^Ω) = ψ0(Ω_2^Ω_2^Ω)
#{(&_2)^(&_2)_1}# has ordinal level ψ0(Ω_2^Ω_2^ψ1(Ω_2)) = ψ0(Ω_2^Ω_2^ε(Ω + 1))
#{(&_2)^(&_2)}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2)
#{(&_2)^(&_2*&_2)}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^2)
#{(&_2)^(&_2)^#}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^ω)
#{(&_2)^(&_2)^#{(&_2)^(&_2)}#}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^ψ0(Ω_2^Ω_2^Ω_2))
#{(&_2)^(&_2)^&}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω)
#{(&_2)^(&_2)^(&_2)_1}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^ψ1(Ω_2)) = ψ0(Ω_2^Ω_2^Ω_2^ε(Ω + 1))
#{(&_2)^(&_2)^(&_2)}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω_2)
#{(&_2)^(&_2)^(&_2)^#}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω_2^ω)
#{(&_2)^(&_2)^(&_2)^&}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω_2^Ω)
#{(&_2)^(&_2)^(&_2)^(&_2)}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
#{(&_2)^(&_2)^(&_2)^(&_2)^(&_2)}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
#{(&_2)^(&_2)^(&_2)^(&_2)^(&_2)^(&_2)}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
#{(&_2)^(&_2)^(&_2)^(&_2)^(&_2)^(&_2)^(&_2)}# has ordinal level ψ0(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
You can see that the &_2 delimiter is analogous to the second uncountable ordinal in the OCF, Ω_2. Let's move on &_3 (note that I am only going to use the Buchholz's function for convenience)...
#{&_3}#[n] = #{&_2^^#}#[n] = #{(&_2)^(&_2)^(&_2)^...^(&_2)^(&_2)^(&_2)}# with n &_2's
#{&_3}# has ordinal level ψ0(Ω_3) = ψ0(ε(Ω_2 + 1))
(#{&_3}#)^^# has ordinal level ψ0(Ω_3 + Ω)
(#{&_3}#)^^## has ordinal level ψ0(Ω_3 + Ω^2)
(#{&_3}#)^^^# has ordinal level ψ0(Ω_3 + Ω^Ω)
(#{&_3}#){&^&}# has ordinal level ψ0(Ω_3 + Ω^Ω^Ω)
(#{&_3}#){(&_2)_1}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2))
(#{&_3}#){(&_2)_1+(&_2)_1}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2)^2)
(#{&_3}#){(&_2)_1*#}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2)^ω)
(#{&_3}#){(&_2)_1*#{&_3}#}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2)^ψ0(Ω_3))
(#{&_3}#){(&_2)_1*&}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2)^Ω)
(#{&_3}#){(&_2)_1*(&_2)_1}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2)^ψ1(Ω_2))
(#{&_3}#){((&_2)_1)^((&_2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2)^ψ1(Ω_2)^ψ1(Ω_2))
(#{&_3}#){(&_2>2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2·2))
(#{&_3}#){(&_2>3)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2·3))
(#{&_3}#){(&_2>#)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2·ω))
(#{&_3}#){(&_2>#{&_3}#)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2·ψ0(Ω_3)))
(#{&_3}#){(&_2>&)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2·Ω))
(#{&_3}#){(&_2>(&_2)_1)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2·ψ1(Ω_2)))
(#{&_3}#){(&_2>&_2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^2))
(#{&_3}#){(&_2>&_2>&_2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^3))
(#{&_3}#){(&_2*#)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^ω))
(#{&_3}#){(&_2*#{&_3}#)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^ψ0(Ω_3)))
(#{&_3}#){(&_2*&)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω))
(#{&_3}#){(&_2*(&_2)_1)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^ψ1(Ω_2)))
(#{&_3}#){(&_2*&_2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω_2))
(#{&_3}#){(&_2^#)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω_2^ω))
(#{&_3}#){(&_2^&)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω_2^Ω))
(#{&_3}#){(&_2^(&_2)_1)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω_2^ψ1(Ω_2)))
(#{&_3}#){(&_2^&_2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω_2^Ω_2))
(#{&_3}#){(&_2^&_2^&_2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω_2^Ω_2^Ω_2))
(#{&_3}#){(&_2^&_2^&_2^&_2)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2))
...
(#{&_3}#){(&_3)_1)}# has ordinal level ψ0(Ω_3 + ψ1(Ω_3))
...
To be continued...