" As we have seen in the previous chapters, Bowers' exploding array function (BEAF) is very powerful and very general, if extremely difficult to define precisely. It is possible however to construct other systems of equal power, and some of these are easier to define than others. Due to the popularity of my Hyper-E and Extended Hyper-E Notation, I have decided to develop a major extension to my "Extensible-E System" (ExE for short) which brings it to the next level of sophistication. One motivation for this is that my Extensible-E System is heavily inspired by my original work with large numbers as a kid, so it's natural that I should develop it further with the hindsight of several years of study. This new extension to Extensible-E is called "Cascading-E Notation" (E^ for short). The reasons for this choice will be made clear as we work out it's details. Before we look into this, I'd first like to address a weak extension to my Extended Hyper-E which was used briefly on the googology wiki (it has since been deleted due to their citation policy). " - Sbiis Saibian
Check out the original source here.
(I am not going to use the "Plus notation")
We will now begin to develop "Cascading-E Notation" (E^). The system begins by re-interpreting a string of hyperions as a product of hyperions. That is:
###...### w/a #s = #*#*#*#* ... *#*# w/a #s = #^a = #a
Note that this gives us a nifty shorthand for writing out certain numbers in Extended Hyper-E:
[I] godgahlah = E100#### ... ####100 w/100 #s = E100#^(100)100
Note that "^" is a new symbol being introduced to our Extensible-E System. Initially the only symbols being used were "E" and "#". We now add the third symbol "^" to the system. This is normally known as the "caret" symbol. For the purposes of Cascading-E Notation it will be called a carrier mark, carrier or carrion. In addition to this we will also be adding the "*" (asterisk) as a punctuation symbol, with the usual multiplicative meaning. Note however that multiplications are not commutative in E^. We will also use "(" and ")" (parentheses) in the usual way to clarify order of operations.
The next step is crucial. Rather than simply allowing us to raise a hyperion to any power, we also allow the exponent to itself be a expression composed of hyperions. The simplest case of this would be "#^#". This becomes our next separator after the infinite set of separators , {#, ##, ###, ####, ...}. We begin with this new definition for the new separator:
@a#^#b = @a#^(b)a = @a### ... ###a w/b #s
As you can see the hyperion within the exponent gets replaced by "b". Furthermore, we can think of this new separator as a generic power of the hyperion:
#^# = ####...####
This notation allows us to write numbers in the godgahlah group as:
godgahlah = E100#^#100
[II] grand godgahlah = E100#^#100#2
[III] grand grand godgahlah = E100#^#100#3
...
[IV] godgahlahgong = E100,000#^#100,000
[V] grand godgahlahgong = E100,000#^#100,000#2
...
[VI] hundred-ex-grand godgahlah = E100#^#100#101
[VI] godgahlah-ex-grand godgahlah = E100#^#100#(godgahlah+1) = E100#^#100#(E100#^#100+1)
As you can see, all the rules of Hyper-E and Extended Hyper-E carry over to the new notation.
We continue by extrapolating a pattern observed with the existing separators within a new "hash-product" paradigm. Observe that every succeeding separator is multiplying the previous separator by a hyperion.
e.g.
#*# = ##
##*# = ###
###*# = ####
#*## = ###
#*### = ####
##*## = ####
etc.
So here we have repetitions of the #^# separator.
Let's go!
#^# is the limit hyperion of {#, ##, ###, ####, ...} (n-copies of #)
#^#*# => Ea#^#*#b = Ea#^#a#^#...#^#a#^#a w/ b a's
#^#*## => Ea#^#*##b = Ea#^#*#a#^#*#...#^#*#a#^#*#a w/ b a's
#^#*#^# is the limit hyperion of {#^#, #^#*#, #^#*##, #^#*###, ...}
#^#*#^#*#^# is the limit hyperion of {#^#*#^#, #^#*#^#*#, #^#*#^#*##, ...}
#^## is the limit hyperion of {#^#, #^#*#^#, #^#*#^#*#^#, ...}
#^##*#^# is the limit hyperion of {#^##, #^##*#, #^##*##, ...}
#^##*#^## is the limit hyperion of {#^##, #^##*#^#, #^##*#^#*#^#, ...}
#^### is the limit hyperion of {#^##, #^##*#^##, #^##*#^##*#^##, ...}
#^#^# is the limit hyperion of {#^#, #^##, #^###, #^####, ...}
#^(#^#*#) is the limit hyperion of {#^#^#, #^#^#*#^#^#, #^#^#*#^#^#*#^#^#, ...}
#^(#^#*##) is the limit hyperion of {#^(#^#*#), #^(#^#*#)*#^(#^#*#), #^(#^#*#)*#^(#^#*#)*#^(#^#*#), ...}
#^(#^#*#^#) is the limit hyperion of {#^#^#, #^(#^#*#), #^(#^#*##), ...}
#^#^## is the limit hyperion of {#^#^#, #^(#^#*#^#), #^(#^#*#^#*#^#), ...}
#^#^#^# is the limit hyperion of {#^#^#, #^#^##, #^#^###, ...}
and finally, #^^# is the limit hyperion of {#, #^#, #^#^#, #^#^#^#, #^#^#^#^#, ...}, making #^^# diagonalizes over the limit of Cascading-E notation.
Furthermore, (#^#)^# = #^##, (#^#^#)^# = #^(#^#*#), (#^#^#)^#^# = #^(#^#*#^#), etc. This because the exponentiation rule, using the same reasoning as the exponentiation rule on integers.
[1] Base rule: E[b]a = b^a
[2] The hyper-product expansion:
If there is a complex hyper-product between the last two arguments whose the last cascader is not of the form ###...###, then Let & be the hyper-product and &'(b) be the transformation with base b; we can then say @a&b = @a&'(b)a
[3] The truncation rule:
If there is a hyper-product between the last two arguments whose the last cascader is of the form ##...## and the last argument is 1, then let & be the hyper-product; we can then say @a&1 = @a
[4] The hyper-band expansion rule:
If the last argument is greater than 1 and the last cascader is of the form #^n where n >= 1, then Let & be the hyper-product and &*# be the hyper-product with one more hyperion at the end; we can then say @a&*#b = @a&a& ... &a&a w/b a's
[5] The recursion rule:
If none of these apply than the last argument must be greater than 1, and the last separator is the proto-hyperion, "#"; we then just say that @a#b = @(@a#(b-1))
Before getting more extraordinary, let's peak on more ecetonates beyond throogol!
Level 1 | Stage 3-6A
Stage 3-6A ~ Revising centillion-based numbers beyond throogol
Category 29 - 44
Regiment block 57
Members: 106
(8022 - 8127)
(8022) eceto-throogol = E303###303
(8023) eceto-throogolplexithrais = E303###303#2
(8024) eceto-throogolduplexithrais = E303###303#3
...
(8025) eceto-thrangol = E303###303#303
(8026) eceto-threagol = E303###303#303#303
(8027) eceto-thrigangol = E303###303#303#303#303
(8028) eceto-throrgegol = E303###303##5
(8029) eceto-thrulgol = E303###303##6
(8030) eceto-thraspgol = E303###303##7
(8031) eceto-thrinorgol = E303###303##8
(8032) eceto-thrargantuul = E303###303##9
(8033) eceto-throogondol = E303###303##10
...
(8034) eceto-thrugold = E303###303##303
(8035) eceto-thraatagold = E303###303##303#303
(8036) eceto-threegold = E303###303##303#303#303
(8037) eceto-thrinningold = E303###303##303#303#303#303
(8038) eceto-throlaagold = E303###303##303##5
(8039) eceto-thruelohgold = E303###303##303##6
(8040) eceto-thraspgold = E303###303##303##7
(8041) eceto-thrinorgold = E303###303##303##8
(8042) eceto-thrargantuuld = E303###303##303##9
(8043) eceto-throogondold = E303###303##303##10
...
(8044) eceto-thrugolthra = E303###303##303##303
(8045) eceto-thraatagolthra = E303###303##303##303#303
(8046) eceto-threegolthra = E303###303##303##303#303#303
...
(8047) eceto-thrugoltesla = E303###303##303##303##303
(8048) eceto-thraatagoltesla = E303###303##303##303##303#303
...
(8049) eceto-thrugolpeta = E303###303##303##303##303##303
(8050) eceto-thrugolhexa = E303###303##303##303##303##303##303
(8051) eceto-thrugolhepta = E303###303###7
(8052) eceto-thrugolocta = E303###303###8
(8053) eceto-thrugolenna = E303###303###9
(8054) eceto-thrugoldeka = E303###303###10
...
(8055) eceto-throotrigol = E303###303###303
(8056) eceto-thrantrigol = E303###303###303#303
(8057) eceto-threatrigol = E303###303###303#303#303
...
(8058) eceto-thrutrigold = E303###303###303##303
(8059) eceto-thraatatrigold = E303###303###303##303#303
(8060) eceto-threetrigold = E303###303###303##303#303#303
...
(8061) eceto-thrutrigolthra = E303###303###303##303##303
(8062) eceto-thraatatrigolthra = E303###303###303##303##303#303
...
(8063) eceto-thrutrigoltesla = E303###303###303##303##303##303
(8064) eceto-thrutrigolpeta = E303###303###303###5
(8065) eceto-thrutrigolhexa = E303###303###303###6
...
(8066) eceto-throotergol = E303###303###303###303
(8067) eceto-thrantergol = E303###303###303###303#303
(8068) eceto-threatergol = E303###303###303###303#303#303
...
(8069) eceto-thrutergold = E303###303###303###303##303
(8070) eceto-thraatatergold = E303###303###303###303##303#303
...
(8071) eceto-thrutergolthra = E303###303###303###303##303##303
(8072) eceto-thrutergoltesla = E303###303###303###303##303##303##303
(8073) eceto-thrutergolpeta = E303###303###303###303###5
(8074) eceto-thrutergolhexa = E303###303###303###303###6
...
(8075) eceto-throopetol = E303###303###303###303###303
(8076) eceto-thranpetol = E303###303###303###303###303#303
(8077) eceto-thrupetold = E303###303###303###303###303##303
(8078) eceto-thrupetolthra = E303###303###303###303###303##303##303
...
(8079) eceto-throohexol = E303###303###303###303###303###303
(8080) eceto-thranhexol = E303###303###303###303###303###303#303
(8081) eceto-thruhexold = E303###303###303###303###303###303##303
(8082) eceto-thruhexolthra = E303###303###303###303###303###303##303##303
...
(8083) eceto-throoheptgol = E303###303###303###303###303###303###303 = E303####7
(8084) eceto-throogogdgol = E303###303###303###303###303###303###303###303 = E303####8
(8085) eceto-throogentol = E303###303###303###303###303###303###303###303###303 = E303####9
(8086) eceto-throodekol = E303###303###303###303###303###303###303###303###303###303 = E303####10
...
(8087) eceto-tetroogol = E303####303
(8088) eceto-tetroogolplexitetrais = E303####303#2
...
(8089) eceto-tetrangol = E303####303#303
(8090) eceto-tetreagol = E303####303#303#303
...
(8091) eceto-tetrugold = E303####303##303
(8092) eceto-tetraatagold = E303####303##303#303
...
(8093) eceto-tetrugolthra = E303####303##303##303
(8094) eceto-tetrugoltesla = E303####303##303##303##303
...
(8095) eceto-tetrithroogol = E303####303###303
(8096) eceto-tetrithrootrigol = E303####303###303###303
(8097) eceto-tetrithrootergol = E303####303###303###303###303
(8098) eceto-tetrithroopetol = E303####303####5
(8099) eceto-tetrithroohexol = E303####303####6
...
(8100) eceto-tetrootrigol = E303####303####303
(8101) eceto-tetratrithroogol = E303####303####303###303
(8102) eceto-tetratrithrootrigol = E303####303####303###303###303
...
(8103) eceto-tetrootergol = E303####303####303####303
(8104) eceto-tetraterithroogol = E303####303####303####303###303
...
(8105) eceto-tetroopetol = E303####303####303####303####303
(8106) eceto-tetroohexol = E303####303####303####303####303####303
(8107) eceto-tetrooheptgol = E303#####7
(8108) eceto-tetroogogdol = E303#####8
(8109) eceto-tetroogentol = E303#####9
(8110) eceto-tetroodekol = E303#####10
...
(8111) eceto-pentoogol = E303#####303
(8112) eceto-pentrangol = E303#####303#303
(8113) eceto-pentugold = E303#####303##303
(8114) eceto-pentithroogol = E303#####303###303
(8115) eceto-pentitetroogol = E303#####303####303
...
(8116) eceto-pentootrigol = E303#####303#####303
(8117) eceto-pentootergol = E303#####303#####303#####303
(8118) eceto-pentoopetol = E303######5
(8119) eceto-pentoohexol = E303######6
...
(8120) eceto-hexoogol = E303######303
(8121) eceto-hexootrigol = E303######303######303
(8122) eceto-hexootergol = E303######303######303######303
...
(8123) eceto-heptoogol = E303#######303
(8124) eceto-ogdoogol = E303########303
(8125) eceto-entoogol = E303#########303
(8126) eceto-dektoogol = E303##########303
...
(8127) eceto-triacositriolus ✡ = E303#^(303)303 = E303#^#303
= E303### ### ### ... ... ... ### ### ###303
w/ 303 #'s
...
What about the beyond? Let's continue the first Cascading-E regiment!