Linear-array extended cascading-E notation

We have got a loooong way to go, so let's get going. 

Array Decomposition Rules of Linear-Array-Extended-Cascading-E Notation ({}xE^)

All rules from xE^ still apply, so products etc. work the same. Here are the rules for decomposing linear array separators. I use letters (a,b,c,...) for individual array entries, % for the rest of the array, $ for a string containing 0 or more entries of "1", and @ is a string of entries each identical to the first entry. BEAF terminology still applies.

(If there are any cases where the definition doesn't work, please let me know on my talk page!)

This notation is expected to have a limit of the SVO in the fast-growing hierarchy in Buchholz's/Madore's psi, assuming everything "works well".

Godsgodgulus Regiment (Revised)

order type 𝜑(w,0,0)


Blasphemorgulus regiment


order type 𝜑(1,0,0,0) (Ackermann ordinal)


Warning: The numbers here are only partially formalised under my conservative interpretation of ExE (which Cookie Fonster uses too. Saibian goes to {#,#+1,1,2}, for example, whilst the same interpretation would yield #{{1}}#># (or {#,#,1,2}>#) under my "climbing" interpretation. I prefer my interpretation for two reasons: the order type is, at least, semi-predictable, and that it goes under the practice of ExE seen in the -cthulhum googolisms.


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Here I've left out all numbers between the order types of 𝜑(1,0,0,0) and 𝜑(1,0,0,1) because Saibian has enough googolisms to fill up this gap. 

On the Googology Wiki page of this regiment there is no need to include these intermediary numbers.

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Blasphemordeugulus regiment


order type 𝜑(1,1,0,0)



Blasphemortrigulus regiment


order type 𝜑(1,2,0,0)



Blasphemortergulus regiment


order type 𝜑(1,3,0,0)



Blasphemorpengulus regiment


order type 𝜑(1,4,0,0)



Blasphemorhexgulus regiment


order type 𝜑(1,5,0,0)



Blasphemorhepgulus regiment


order type 𝜑(1,6,0,0)



Blasphemorattgulus regiment


order type 𝜑(1,7,0,0)



Blasphemorenngulus regiment


order type 𝜑(1,8,0,0)



Blasphemordekgulus regiment


order type 𝜑(1,9,0,0)



Blasphegodgulus regiment


order type 𝜑(1,w,0,0)



Blasphemorguldeus regiment


order type 𝜑(2,0,0,0)



Blasphemorgultruce regiment


order type 𝜑(3,0,0,0)



Blasphemorgulquad regiment


order type 𝜑(4,0,0,0)



Godsgophemorgulus regiment


order type 𝜑(w,0,0,0)



Pentemorgulus regiment


order type 𝜑(1,0,0,0,0)



Hexemorgulus regiment


order type 𝜑(1,0,0,0,0,0)



Heptemorgulus regiment


order type 𝜑(1,0,0,0,0,0,0)



Ottemorgulus regiment


order type 𝜑(1,0,0,0,0,0,0,0)



Entemorgulus regiment


order type 𝜑(1,0,0,0,0,0,0,0,0)



Deckemorgulus regiment


order type 𝜑(1,0,0,0,0,0,0,0,0,0)



Ominongulus regiment & way way beyond (BETA)


order type ψ(ΩΩ^w) (small veblen ordinal)

I put this in beta because this regiment will likely be never be completed in finite time. As we can continue using BEAF structures to form the array in ##xE^, there may be so many more numbers I could coin that I most likely will never bother coining.

Now we will use the -odgulus suffix to represent {#,n(1)2}.

hendektodgulus = E100{#,11(1)2}100

dodektodgulus = E100{#,12(1)2}100

trisdektodgulus = E100{#,13(1)2}100

terdektodgulus = E100{#,14(1)2}100

pendektodgulus = E100{#,15(1)2}100

exdektodgulus = E100{#,16(1)2}100

eptdektodgulus = E100{#,17(1)2}100

ogddektodgulus = E100{#,18(1)2}100

entdektodgulus = E100{#,19(1)2}100

icosodgulus = E100{#,20(1)2}100

triantodgulus = E100{#,30(1)2}100

sarantodgulus = E100{#,40(1)2}100

penintodgulus = E100{#,50(1)2}100

exintodgulus = E100{#,60(1)2}100

ebdomintodgulus = E100{#,70(1)2}100

ogdondodgulus = E100{#,80(1)2}100

eninodgulus = E100{#,90(1)2}100

ominongulus/hectodgulus = E100{#,#(1)2}100 = E100{#,2,1,1,... ...,1,2} a.p. #+1

gigasodgulus = E100{#,#(1)2}500

chiliasodgulus = E100{#,#(1)2}1,000

myriasodgulus = E100{#,#(1)2}10,000

hotalodgulus = E100{#,#(1)2}100,000

pseudomonarchia daemonum = E100{#,#(1)2}44435622

octadodgulus = E100{#,#(1)2}100,000,000

popiadodgulus = E100{#,P(1)2}100

sedeniadodgulus = E100{#,#(1)2}10,000,000,000,000,000

googlodgulus = E100{#,#(1)2}(E100)

granglodgulus = E100{#,#(1)2}(E100#100)

clarkkkksonglodgulus = E100{#,¥(1)2}100 

godgahlodgulus = E100{#,#(1)2}(E100#^#100)

godsgodgulodgulus = E100{#,#(1)2}(E100#{#}#100)

terodgodgulus = E100{#,#(1)2}(E100{#,4(1)2}100)

grand ominongulus = E100{#,#(1)2}100#2 = E100{#,ominongulus(1)2}100

grand pseudomonarchia daemonum = E100{#,#(1)2}44435622#2

grand grand ominongulus = E100{#,#(1)2}100#3

grand grand grand ominongulus = E100{#,#(1)2}100#4

grangol-carta-ominongulus = E100{#,#(1)2}100#100

greagol-carta-ominongulus = E100{#,#(1)2}100#100#100

gigangol-carta-ominongulus = E100{#,#(1)2}100#100#100#100

gugolda-carta-ominongulus = E100{#,#(1)2}100##100

throogol-carta-ominongulus = E100{#,#(1)2}100###100

godgahlah-carta-ominongulus = E100{#,#(1)2}100#^#100

godsgodgulus-carta-ominongulus = E100{#,#(1)2}100#{#}#100

blasphemorgulus-carta-ominongulus = E100{#,#(1)2}100#{{1}}#100

pententrimorgulus-carta-ominongulus = E100{#,#(1)2}100{#,#,1,1,2}100

hexentrimorgulus-carta-ominongulus = E100{#,#(1)2}100{#,#,1,1,1,2}100

heptentrimorgulus-carta-ominongulus = E100{#,#(1)2}100{#,#,1,1,1,1,2}100

octentrimorgulus-carta-ominongulus = E100{#,#(1)2}100{#,#,1,1,1,1,1,2}100

ententrimorgulus-carta-ominongulus = E100{#,#(1)2}100{#,#,1,1,1,1,1,1,2}100

dekentrimorgulus-carta-ominongulus = E100{#,#(1)2}100{#,#,1,1,1,1,1,1,1,2}100

ominongulus-by-deuteron = E100{#,#(1)2}100{#,#(1)2}100

ominongulus-by-triton = E100{#,#(1)2}100{#,#(1)2}100{#,#(1)2}100

ominongulus-by-teterton = E100{#,#(1)2}*#5

ominongulus-by-pepton = E100{#,#(1)2}*#6

ominongulus-by-exton = E100{#,#(1)2}*#7

ominongulus-by-epton = E100{#,#(1)2}*#8

ominongulus-by-ogdon = E100{#,#(1)2}*#9

ominongulus-by-enton = E100{#,#(1)2}*#10

ominongulus-by-dekaton = E100{#,#(1)2}*#11

ominongulus-by-gugold = E100{#,#(1)2}*#100

ominongulus-by-throogol = E100{#,#(1)2}*##100

ominongulus-by-tetroogol = E100{#,#(1)2}*###100

ominongulus-by-pentoogol = E100{#,#(1)2}*####100

ominongulus-by-godgahlah = E100{#,#(1)2}*#^#100

ominongulus-by-godsgodgulus = E100{#,#(1)2}*#{#}#100

ominongulus-by-blasphemorgulus = E100{#,#(1)2}*#{{1}}#100

ominongulus-by-pententrimorgulus = E100{#,#(1)2}*{#,#,1,1,2}100

ominongulus-by-hexentrimorgulus = E100{#,#(1)2}*{#,#,1,1,1,2}100

deutero-ominongulus = E100{#,#(1)2}*{#,#(1)2}100

trito-ominongulus = E100{#,#(1)2}*{#,#(1)2}*{#,#(1)2}100

teterto-ominongulus = E100{#,#(1)2}^#4

pepto-ominongulus = E100{#,#(1)2}^#5

exto-ominongulus = E100{#,#(1)2}^#6

epto-ominongulus = E100{#,#(1)2}^#7

ogdo-ominongulus = E100{#,#(1)2}^#8

ento-ominongulus = E100{#,#(1)2}^#9

dekato-ominongulus = E100{#,#(1)2}^#10

hecato-ominongulus/ominongulfact = E100{#,#(1)2}^#100

grideuterominongulus = E100{#,#(1)2}^##100

kubiku-ominongulus = E100{#,#(1)2}^###100

quarticu-ominongulus = E100{#,#(1)2}^####100

quinticu-ominongulus = E100{#,#(1)2}^#^#5

ominongulipso-godgahlah/centicu-ominongulus = E100{#,#(1)2}^#^#100

ominongulipso-godsgodgulus = E100{#,#(1)2}^#{#}#100

ominongulipso-blasphemorgulus = E100{#,#(1)2}^#{{1}}#100

ominongulipso-pentemorgulus = E100{#,#(1)2}^{#,#,1,1,2}100

ominongulipso-hexemorgulus = E100{#,#(1)2}^{#,#,1,1,1,2}100

dutetrated-ominongulus = E100{#,#(1)2}^{#,#(1)2}100

tritetrated ominongulus = E100{#,#(1)2}^{#,#(1)2}^{#,#(1)2}100

quadratetrated ominongulus = E100{#,#(1)2}^^#4

quinquatetrated ominongulus = E100{#,#(1)2}^^#5

sexatetrated ominongulus = E100{#,#(1)2}^^#6

septatetrated ominongulus = E100{#,#(1)2}^^#7

octatetrated ominongulus = E100{#,#(1)2}^^#8

nonatetrated ominongulus = E100{#,#(1)2}^^#9

decatetrated ominongulus = E100{#,#(1)2}^^#10

terrible ominongulus = E100{#,#(1)2}^^#100

terrisquared ominongulus = E100{#,#(1)2}^^##100

terricubed ominongulus = E100{#,#(1)2}^^###100

territesserated ominongulus = E100{#,#(1)2}^^####100

territoped ominongulus = E100{#,#(1)2}^^#^#100

ominongultetripso-tethrathoth = E100{#,#(1)2}^^#^^#100

ominongultetripso-godsgodgulus = E100{#,#(1)2}^^#{#}#100

ominongultetripso-blasphemorgulus = E100{#,#(1)2}^^#{{1}}#100

ominongultetripso-pententrimorgulus = E100{#,#(1)2}^^{#,#,1,1,2}100

ominongultetripso-hexentrimorgulus = E100{#,#(1)2}^^{#,#,1,1,1,2}100

dupentated-ominongulus = E100{#,#(1)2}^^{#,#(1)2}100

tripentated-ominongulus = E100{#,#(1)2}^^{#,#(1)2}^^{#,#(1)2}100

quadrapentated-ominongulus = E100{#,#(1)2}^^{#,#(1)2}^^{#,#(1)2}^^{#,#(1)2}100

quinquapentated-ominongulus = E100{#,#(1)2}^^^#5

sexapentated-ominongulus = E100{#,#(1)2}^^^#6

septapentated-ominongulus = E100{#,#(1)2}^^^#7

octapentated-ominongulus = E100{#,#(1)2}^^^#8

nonapentated-ominongulus = E100{#,#(1)2}^^^#9

decapentated-ominongulus = E100{#,#(1)2}^^^#10

horrible ominongulus = E100{#,#(1)2}^^^#100

horrendous ominongulus = E100{#,#(1)2}^^^^#100

heptorrendous ominongulus = E100{#,#(1)2}^^^^^#100

ogdorrendous ominongulus = E100{#,#(1)2}{6}#100

ennorrendous ominongulus = E100{#,#(1)2}{7}#100

dekorrendous ominongulus = E100{#,#(1)2}{8}#100

goliathed-ominongulus = E100{#,#(1)2}{10}#100

godsgodgulated-ominongulus = E100{#,#(1)2}{#}#100

poporrendous ominongulus = E100{#,#(1)2}{P}#100 

ohmygosh-ohmygosh-ohmygoooshed-ominongulus = E100{#,#(1)2}{#{#}#}#100

blasphemorgulnumus ominongulus = E100{#,#(1)2}{#{{1}}#}100

duexpanded-ominongulus / ominongulohmygosh-ominongulohmygosh = E100{#,#(1)2}{{#,#(1)2}}{#,#(1)2}100

triexpanded-ominongulus / ominongulohmygosh-ominongulohmygosh-ominongulohmygooosh = E100{#,#(1)2}{{1}}#3

tetrexpanded-ominongulus = E100{#,#(1)2}{{1}}#4

pentexpanded-ominongulus = E100{#,#(1)2}{{1}}#5

blasphemorgulated-ominongulus = E100{#,#(1)2}{{1}}#100

pententrimorgulated-ominongulus = E100{{#,#(1)2},#,1,1,2}100

hexentrimorgulated-ominongulus = E100{{#,#(1)2},#,1,1,1,2}100

heptentrimorgulated-ominongulus = E100{{#,#(1)2},#,1,1,1,1,2}100

octentrimorgulated-ominongulus = E100{{#,#(1)2},#,1,1,1,1,1,2}100

ententrimorgulated-ominongulus = E100{{#,#(1)2},#,1,1,1,1,1,1,2}100

dekentrimorgulated-ominongulus = E100{{#,#(1)2},#,1,1,1,1,1,1,1,2}100

ominondeugulus = E100{{#,#(1)2},#(1)2}100

ominontrigulus = E100{{{#,#(1)2},#(1)2},#(1)2}100

ominonquargulus = E100{#,#(1)2}>#4

ominonquingulus = E100{#,#(1)2}>#5

ominonsexgulus = E100{#,#(1)2}>#6

ominonsepgulus = E100{#,#(1)2}>#7

ominoncodgulus = E100{#,#(1)2}>#8

ominonbrossgulus = E100{#,#(1)2}>#9

ominontemngulus = E100{#,#(1)2}>#10

ominonguliterator = E100{#,#(1)2}>#100

grand ominongulus / great and ominous ominongulus = E100{#,#(1)2}>#100#2

dustaculated-ominongulus = E100{#,#(1)2}>{#,#(1)2}100

tristaculated-ominongulus = E100{#,#(1)2}>{#,#(1)2}>{#,#(1)2}100

tetrastaculated-ominongulus = E100{#,#(1)2}>{#,#(1)2}>{#,#(1)2}>{#,#(1)2}100 = E100{#,##|2}4

pentastaculated-ominongulus = E100{#,##|2}5

hexastaculated-ominongulus = E100{#,##|2}6

hectastaculated-ominongulus = E100{#,##|2}7

ogdastaculated-ominongulus = E100{#,##|2}8

ennastaculated-ominongulus = E100{#,##|2}9

dekastaculated-ominongulus = E100{#,##|2}10

ominongulcross = E100{#,##|2}100

ominongulcubor = E100{#,###|2}100

ominongulteron = E100{#,####|2}100

ominongulpeton = E100{#,#####|2}100

ominongulhexon = E100{#,#^6|2}100

ominongulhepton = E100{#,#^7|2}100

ominongulogdon = E100{#,#^8|2}100

ominongulennon = E100{#,#^9|2}100

ominonguldekon = E100{#,#^10|2}100

ominongultope = E100{#,#^#|2}100

ominongularxitri = E100{#,{#,#|2}|2}100

ominongularxitet = E100{#,{#,{#,#|2}|2}|2}100

ominongularxipent = E100{#,5,2|2}100

ominongularxihex = E100{#,6,2|2}100

ominongularxihept = E100{#,7,2|2}100

ominongularxiogd = E100{#,8,2|2}100

ominongularxienn = E100{#,9,2|2}100

ominongularxideck = E100{#,10,2|2}100

ominontethrathoth / ominondicthulhum / ominongularxihect = E100{#,#,2|2}100

ominongulhen = E100{#,#+1(1)2}100 = E100{#,#|1,2}100

ominonguldis = E100{#,#+2(1)2}100

ominongultris = E100{#,#+3(1)2}100

ominongultet = E100{#,#+4(1)2}100

ominondeus = E100{#,#+#(1)2}100

ominongridulus = E100{#,##(1)2}100

hyperioduodgulus = E100{#,#,2(1)2}100

hyperiotriodgulus = E100{#,#,3(1)2}100

hundrithregulus = E100{#,#,#(1)2}100

hundritetrogulus = E100{#,#,#,#(1)2}100

tritonchirechelon = E100{#,#(1)3}100

tetertonchirechelon = E100{#,#(1)4}100

hecatonchirechelon = E100{#,#(1)#}100

dubogogulus = E100{#,#(1)(1)2}100

triubogogulus = E100{#,#(1)(1)(1)2}100

goxxogogulus = E100{#,#(2)2}100

cloxxogogulus = E100{#,#(3)2}100

teroxxogogulus = E100{#,#(4)2}100

petoxxogogulus = E100{#,#(5)2}100

brobdingnagongule = E100{#,#(0,1)2}100 = E100{X^#&#}100

lugubrigoth = E100{X^^X&#}100

lugridigoth = E100{X^^##&#}100

lucubigoth = E100{X^^###&#}100

lutopogoth = E100{X^^#^#&#}100

kekrakunge = E100{X^^^#&#}100

lugublagoth = E100{X^^X&X&#}100

sprach zarathustra = E100{#,#/2}100

grand sprach zarathustra = E100{#,#/2}100#2

grangol-carta-sprach-zarathustra = E100{#,#/2}100#100

sprach-zarathustri = E100{#,#/2}100{#,#/2}100

sprach-zarathustet = E100{#,#/2}100{#,#/2}100{#,#/2}100

sprach-zarathustra-by-hyperion = E100{#,#/2}*#100

deutero-sprach-zarathustra = E100{#,#/2}*{#,#/2}100

trito-sprach-zarathustra = E100{#,#/2}*{#,#/2}*{#,#/2}100

teterto-sprach-zarathustra = E100{#,#/2}*{#,#/2}*{#,#/2}*{#,#/2}100

sprach-zarathusfact = E100{#,#/2}^#100

sprach-zarathustrifact = E100{#,#/2}^#100{#,#/2}^#100

terrible-sprach-zarathustra = E100{#,#/2}^^#100

horrible-sprach-zarathustra = E100{#,#/2}^^^#100

horrendous-sprach-zarathustra = E100{#,#/2}^^^^#100

sprach-deuter-zarathustra = E100{{#,#/2},#/2}100

sprach-triter-zarathustra = E100{{{#,#/2},#/2},#/2}100

sprach-iterator-zarathustra = E100{#,#/2}>#100

sprach-diterator-zarathustra = E100{#,#/2}>(#+#)100

sprach-griditerator-zarathustra = E100{#,#/2}>##100

sprach-spatialator-zarathustra/godgahlah-turreted-sprach-zarathustra = E100{#,#/2}>#^#100

dustaculated-sprach-zarathustra = E100{#,#/2}>{#,#/2}100

tristaculated-sprach-zarathustra = E100{#,#/2}>{#,#/2}>{#,#/2}100

hectastaculated-sprach-zarathustra/sprach-square-zarathustra = E100{#,#/2}>{#,#/2}>... ... ... ...>{#,#/2}100


Now I use buchholz psi

sprach-cube-zarathustra, approx ψ_0((Ω_ω)Ω^2)

sprach-tesseract-zarathustra, approx ψ_0((Ω_ω)Ω^3)

sprach-tope-zarathustra, approx ψ_0((Ω_ω)Ω^ω)

sprach-zarathustrarxihect, approx ψ_0((Ω_ω)Ω^Ω)

sprach-zarathusthenus, E100{#,#+1/2}100

hyperiated-disprach = E100{#,#,2/2}100

E100{#,#,#/2}100

E100{#,#(1)2/2}100

E100{X^^X&#/2}100

E100{#,#/3}100

E100{#,#/#}100

E100{#,#/1,2}100

E100{#,#/1/2}100

E100{#,#(/1)2}100

E100{#,#(/0,1)2}100

E100{X^^X&&#}100

tonsprach = E100{#,#//2}100

thrinsprach = E100{#,#///2}100

tetrinsprach = E100{#,#////2}100

thus sprach zarathustra = E100{#,#(1)/2}100


... the end??? ...


MORE COMING SOON


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