Check out the source for the Aarex Tiaokhiao's Expand-E notation here!
I decided to compare with Aarex's Expand-E notation that use arrays instead of ampersands. This is extraordinary, since Aarex decided to use arrays in order to continue the Saibian's Hyper-Hyper-Extended Cascading-E notation that Saibian concluded the definition on {#,#,1,2} (now defined as #{{1}}# or #{1,2}#), using Bowers' theory of BEAF structures.
He however, defined with different fundamental sequences of (#{{1}}#){{1}}#, which satisfy some weird fundamental sequences that are unfamiliar with the fundamental sequences associated with the extended Veblen hierarchy or Buchholz's hierarchy, known as "stupid fundamental sequence" (stupid FS), in that case:
(#{{1}}#){{1}}#[1] = #{{1}}# => ordinal level φ(1,0,0,0)
(#{{1}}#){{1}}#[2] = (#{{1}}#){#{{1}}#}(#{{1}}#) => ordinal level φ(φ(1,0,0,0),φ(1,0,0,0),0)
(#{{1}}#){{1}}#[n] = (#{{1}}#){(#{{1}}#){{1}}#[n-1]}(#{{1}}#) => ordinal level φ(φ(φ(1,0,0,0),φ(1,0,0,0),0),φ(1,0,0,0),0)
While Madom's Collapsing-E notation's (#{&}#){&}# is defined as:
(#{&}#){&}#[1] = #{&}# => ordinal level φ(1,0,0,0)
(#{&}#){&}#[2] = (#{&}#){#{&}#}# => ordinal level φ(φ(1,0,0,0),0,1)
(#{&}#){&}#[n] = (#{&}#){(#{&}#){&}#[n-1]}# => ordinal level φ(φ(φ(1,0,0,0),0,1),0,0)
Moreover, Aarex Expand-E fundamental sequence of "diabhalegotoaa" series, satisfies a similar "stupid fundamental sequence" as follows:
[A] diabhalegotoaa = E100#[{1}]#100
"diabhal" (devil in Irish) + "megoton" (mine)
With the fundamental sequence:
#[{1}]#[1] = # => ordinal level 1 (as in grangol = E100#100)
#[{1}]#[2] = #{#}^(#)# => ordinal level φ(ω,ω,0,0)
#[{1}]#[3] = #{#}^(#{#}^(#)#)#)# => ordinal level φ(φ(ω,ω,0,0),ω,0,0)
#[{1}]#[n] = #{#}^(#[{1}]#[n-1])# => ordinal level φ(1,0,0,0,0)
The array system, however, this is not compatible with the first uncountable cardinal of the Buchholz's OCF. In that case, I decided to choose the ampersand symbol rather than arrays inside curly brackets for convenience, and to make the function of the system much more powerful!
For instance, Aarex's Expand-E notation is defined using distinctive brackets based off BEAF, that is hard to read and understand, while Madom's Collapsing-E notation is much easier! While Aarex's Expand-E notation only reaches #{&&}#-level of the Collapsing-E (equivalent to ordinal level φ(1,0,0,0,0) using extended Veblen hierarchy, or ψ0(Ω^Ω^3) using Buchholz's function in the fast-growing hierarchy), although Aarex did not define the extension of the ExE further, is ill-defined beyond that level, and fails to reach the limit of the Collapsing-E. Madom's Collapsing-E notation is more well-defined, much stronger, and even reaches the Bachmann-Howard ordinal level (equivalent to ψ0(Ω_2) using Buchholz's function) in the FGH!
#{{1}}# in E<> is exactly equal to #{&}# in &E, and have ordinal level φ(1,0,0,0)
(#{{1}}#)^^# in E<> = (#{&}#)^^# in &E, and have ordinal level ε(φ(1,0,0,0)+1)
(#{{1}}#)^^^# in E<> = (#{&}#)^^^# in &E, and have ordinal level Γ(φ(1,0,0,0)+1)
(#{{1}}#){#}# in E<> = (#{&}#){#}# in &E, and have ordinal level φ(ω,0,φ(1,0,0,0)+1)
(#{{1}}#){#{{1}}#}# in E<> = (#{&}#){#{&}#}# in &E, and have ordinal level φ(φ(1,0,0,0),0,1)
(#{{1}}#){#{{1}}#}(#{{1}}#) in E<> = (#{&}#){#{&}#}(#{&}#) in &E, and have ordinal level φ(φ(1,0,0,0),φ(1,0,0,0),0)
On the other hand...
(#{{1}}#){{1}}# in E<> and (#{&}#){&}# have the same ordinal level of φ(1,0,0,1), but with different fundamental sequences, so E<>'s (#{{1}}#){{1}}# > &E's (#{&}#){&}#.
Moving on...
E<>'s #{{2}}# > &E's #{&+1}# => ordinal level φ(1,1,0,0)
E<>'s #{{3}}# > &E's #{&+2}# => ordinal level φ(1,2,0,0)
E<>'s #{{4}}# > &E's #{&+3}# => ordinal level φ(1,3,0,0)
E<>'s #{{#}}# < &E's #{&+#}# => ordinal level φ(1,ω,0,0)
E<>'s #{{#+1}}# < &E's #{&+#+1}# => ordinal level φ(1,ω+1,0,0)
E<>'s #{{#+#}}# < &E's #{&+#+#}# => ordinal level φ(1,ω2,0,0)
E<>'s #{{##}}# < &E's #{&+##}# => ordinal level φ(1,ω^2,0,0)
E<>'s #{{#^#}}# < &E's #{&+#^#}# => ordinal level φ(1,ω^ω,0,0)
E<>'s #{{#^^#}}# < &E's #{&+#^^#}# => ordinal level φ(1,ε0,0,0)
E<>'s #{{#{{1}}#}}# < &E's #{&+#{&}#}# => ordinal level φ(1,φ(1,0,0,0),0,0)
E<>'s #{{#{{#}}#}}# < &E's #{&+#{&+#}#}# => ordinal level φ(1,φ(1,ω,0,0),0,0)
E<>'s #{{{1}}}# < &E's #{&+&}# => ordinal level φ(2,0,0,0)
E<>'s #{{{2}}}# > &E's #{&+&+1}# => ordinal level φ(2,1,0,0)
E<>'s #{{{3}}}# > &E's #{&+&+2}# => ordinal level φ(2,2,0,0)
E<>'s #{{{#}}}# < &E's #{&+&+#}# => ordinal level φ(2,ω,0,0)
E<>'s #{{{{1}}}}# < &E's #{&+&+&}# => ordinal level φ(3,0,0,0)
E<>'s #{{{{#}}}}# < &E's #{&+&+&+#}# => ordinal level φ(3,ω,0,0)
E<>'s #{{{{{1}}}}}# < &E's #{&+&+&+&}# => ordinal level φ(4,0,0,0)
E<>'s #{{{{{#}}}}}# < &E's #{&+&+&+&+#}# => ordinal level φ(4,ω,0,0)
E<>'s #{{{{{{1}}}}}}# < &E's #{&+&+&+&+&}# => ordinal level φ(5,0,0,0)
E<>'s #{{{{{{{1}}}}}}}# < &E's #{&+&+&+&+&+&}# => ordinal level φ(6,0,0,0)
E<>'s #{1}^(#)# < &E's #{&#}# => ordinal level φ(ω,0,0,0)
E<>'s #{2}^(#)# > &E's #{&#+1}# => ordinal level φ(ω,1,0,0)
E<>'s #{3}^(#)# > &E's #{&#+2}# => ordinal level φ(ω,2,0,0)
E<>'s #{#}^(#)# < &E's #{&#+#}# => ordinal level φ(ω,ω,0,0)
E<>'s #{#}^(#+1)# > &E's #{&#+&}# => ordinal level φ(ω+1,ω,0,0) (E<>) vs φ(ω+1,0,0,0) (&E)
...
Aarex's concluded at:
E<>'s #[{1}]# is less than &E's #{&&}#, but having the same ordinal level (φ(1,0,0,0,0) using Veblen function) in the fast-growing hierarchy.
With different fundamental sequences:
#{&&}#[1] = #{&}#
#{&&}#[2] = #{&#{&}#}#
#{&&}#[n] = #{&#{&&}#[n-1]}#
Moving on something ill-defined, such as:
#{1,1,2}# = #[{1}]# < #{&&}#
#{2,1,2}# = #[{2}]# > #{&&+1}#
#{#,1,2}# = #[{#}]# < #{&&+#}#
#{1,2,2}# = #[{{1}}]# < #{&&+&}
#{#,2,2}# = #[{{#}}]# < #{&&+&+#}#
#{1,3,2}# = #[{{{1}}}]# < #{&&+&+&}#
#{1,#,2} = #[{1}^#]# < #{&&+&#}#
#{#,#,2}# = #[{#}^#]# < #{&&+&#+#}#
#{1,1,3}# = #[[{1}]]# < #{&&+&&}#
#{1,1,4}# = #[[[{1}]]]# < #{&&+&&+&&}#
#{1,1,#}# = #[{1}]^(#)# < #{&&#}#
#{#,#,#}# = #[{#}^#]^(#)# < #{&&#+&#+#}#
#{1,1,1,2}# < #{&&&}#
#{1,1,1,3}# < #{&&&+&&&}#
#{#,#,#,#}# < #{&&&#+&&#+&#+#}#
#{1,1,1,1,2}# < #{&&&&}#
#{1,1,1,1,1,2}# < #{&&&&&}#
#{1,1,1,1,1,1,2}# < #{&&&&&&}#
But why...
FS of #{#(1)2}# < #{&^#}#
FS of #{1,2(1)2}# < #{&^&}#
With an undefined partner:
[2] diabhalegotoxx ?
“
Transfinite Bowers' Hyperoperators and 4-entry arrays are just the beginning. I have sealed it off from the unholy stargate and defined it in an ExE way.
Expand-E Notation defines these arrays by an existing climbing theory! There's no need to use another hyperoperator notation like Conway's chained arrow notation.
People might know what #{{1}}#+2 is, as they don't mind trouble with hyperoperators when it comes with post-tetrational arrays.
Formalizing the climbing theory is not that hard for BEAF masters. You can even extend it with linear arrays!
Now for a question for future endeavors: #[{1}]## is equal to... You know what to do!
Also check out the Linear Array-E Notation by Username5243 if you want to go further (although it's not formalized).
There's not far away from the next roadblock, where the climbing method stops working. Are we heading for future extensions or stumble at {#,#+1(1)2}, or #{1(1)2}#?
"Say that you are on the go. There's no trouble that holds you from the go. Ooo-oh."
”
Oh, check out the extra stuff I recommended reading, if you are curious. They extend the possibilities of ExE number coding range.
NEXT - Sneak peak #5 - Check out for the list of numbers (beta)