Let's start with where we left off.
Jump to different milestones on this page (due to how google sites's links work it will open in a new tab):
{10,10/2} ~ f_ψ(Ω_ω) (10) ~ s(10,10{1,,1,2}2) ~ (0,0,0)(1,1,1)[10] ~ Y(1,2,4,8)[10]
This entry serves as a "discussion point" for the ordinal, ψ(Ω_ω) and the new level we reached.
We reached limits of many notations. for example Pair Sequence System has been obliterated!
We reached Y(1,2,4,8) in the Y-sequence, (0,0,0)(1,1,1) in BMS, {1,,1,2} separators in SAN, and a whole new level of BAN and BEAF!!
Anyway we have to head on to...
Triakulusplex = {3,3,2/2} = {3,{3,3/2}/2}
The Whipper = {10,100,2/2}
Big boowa = {3,3,3/2} = {3,{3,3,2/2},2/2}
The Whapper = {10,100,3/2}
Great boowa = {3,3,4/2} = {3,{3,3,3/2},3/2}
The Wheeper = {10,100,4/2}
The Whowper = {10,100,5/2}
The Whupper = {10,100,6/2}
The Whiper = {10,100,7/2}
The Whaper = {10,100,8/2}
The Whouper = {10,100,9/2}
Tridecotwix = {10,10,10/2}
The Whurper = {10,100,10/2}
The Whoiper = {10,100,11/2}
The Whuuper = {10,100,12/2}
The Whepper = {10,100,13/2}
The Wheerper = {10,100,14/2}
Forcongulus = {10,10,100/2}
Gugold-carta-Sprach Zarathusthra = E100{#,#/2}100##100
Grand boowa = {3,3,1,2/2} = {3,3,{3,3,3/2}/2}
Big biwwa = {3,3,3,2/2}
Big bawwa = {3,3,3,3/2}
Big beewa = {3,3,3,4/2}
Big biwa = {3,3,3,5/2}
Big bowwa = {3,3,3,6/2}
Big bawa = {3,3,3,7/2}
Generotwix = {10,10,10,10/2}
Supromulus = {10,10,10,100/2}
Throogol-carta-Sprach Zarathusthra = E100{#,#/2}100###100
Suprimulus = {10,10,10,100,2/2}
Supramulus = {10,10,10,100,3/2}
Supreemulus = {10,10,10,100,4/2}
Pentadecotwix = {10,10,10,10,10/2}
Terpromulus = {10,10,10,10,100/2}
Hexadecotwix = {10,10,10,10,10,10/2}
Peppromulus = {10,10,10,10,10,100/2}
Iterotwix = {10,10(1)2/2}
Goobiturbos = {10,100(1)2/2}
Godgahlah-carta-Sprach Zarathusthra = E100{#,#/2}100#^#100
Xappamonga = {10,10(2)2/2}
Gridgahlah-carta-Sprach Zarathusthra = E100{#,#/2}100#^##100
Colossamonga = {10,10(3)2/2}
Kubikahlah-carta-Sprach Zarathusthra = E100{#,#/2}100#^###100
Super Gongulus = {10,10(100)2/2}
Godgathor-carta-Sprach Zarathusthra = E100{#,#/2}100#^#^#100
And so on...
Sprach Zarathusthra-by-hyperion = E100{#,#/2}*#100
Wompogulus = {10,10(10)2/100}
Goobetowber = {10,100(/1)2}
Hectastaculated-Sprach Zarathusthra = E100{#,#/2;#,##}100 (in Punctuation-Extended Cascading-E)
Maybe? Maybe not. Maybe? Maybe not. Hey, we're not here to say that.
It will become increasingly hard to notate Extensible-E above here with more and more impases.
So let's discard Extensible-E. anyway Sbiis hasn't even fully formalized tetrentrical arrays (below generatrix) yet.
Goobetowtrer = {10,100(/1)3}
Goobetowquadrer = {10,100(/1)4}
Gossicrebidator = {10,10(/1)100}
Xapwogulus = {10,10(/2)2} = 10^2 && 10
Goxitorg = {10,100(/2)2} = 100^2 && 10
Goxiborg = {10,100(/2)(/2)2} = 100^2+100^2 && 10
Closswogulus = {10,10(/3)2} = 10^3 && 10
Guapamonga = {10,10(/100)2} = 10^100 && 10
mmm... some delicious guapamonga.
Promtrilgelasphigolempularxile = {12,783(/81529572)11}
this number was surely made for fun.
Ginguamonga = {10,100(/0,2)2} = 10^(100*2) && 10
Ganguamonga = {10,100(/0,3)2} = 10^(100*3) && 10
Geenguamonga = {10,100(/0,4)2} = 10^(100*4) && 10
Boppamonga = {10,100(/0,0,1)2} = 10^(10*100) && 10
Troppamonga = {10,100(/0,0,0,1)2} = 10^(10*10*100) && 10
Goplamonga = {10,100(/(1)1)2} = 10^10^100 && 10
Goduplamonga = {10,100(/(0,1)1)2} = 10^10^10^100 && 10
First kalersity = 100^^100 && 10
10^^^10 && 10
10^^^^10 && 10
{10,10,5} && 10
{10,10,10} && 10
Second kalersity = 100{{1}}3 && 10 = 100{100{100}100}100 && 10
{10,10,{10,10,{10,10,10}}} && 10
{10,5,1,2} && 10
{10,10,1,2} && 10
{10,10(1)2} && 10
Goobemilbliss = {10,100(1)2} && 10 = 100 & 10 && 10
10 & 10 & 10 && 10
{10,4/2} && 10
{10,5/2} && 10
{10,10/2} && 10
Even {X,X,2/2} & 10 is bigger, and is at ψ(Ω_Ω) in the fast-growing hierarchy.
However, in the slow-growing hierarchy, a "whipper" would be {10,100,2/2} in BEAF, ψ(Ω_Ω).
As you can see, the fast-growing hierarchy is the array structure of the slow-growing hierarchy,
like epsilon_0 = ψ(Ω) in the fast-growing hierarchy reaches Bachmann-Howard Ordinal = ψ(Ω_2) in the slow-growing hierarchy.
Same reason that both of them hit ψ(Ω_ω) at the place where we start part 7 (this part!!)
X^^X | ({X,X/2} + 1) && 10
X^^X^X | ({X,X/2} + 1) && 10
X^^^X | ({X,X/2} + 1) && 10
X^^^^X | ({X,X/2} + 1) && 10
X{X}X | ({X,X/2} + 1) && 10
X{{1}}X | ({X,X/2} + 1) && 10
{X,X(1)2} | ({X,X/2} + 1) && 10
{X,X(0,1)2} | ({X,X/2} + 1) && 10
{X,X((1)1)2} | ({X,X/2} + 1) && 10
{X^^X&X} | ({X,X/2} + 1) && 10
... ... ... ... ... ...
X_X | ... X_4 & X_3 & X_2 & X & 10 ~ {X,X+1/2} & 10
X_X^^X | ... X_4 & X_3 & X_2 & X & 10
Buchholz' OCF gets stuck here. also this ordinal is also known as "Takeuti-Feferman-Buchholz Ordinal", or TFBO for short.
So we will use Extended Buchholz' OCF, where this would be psi0(Omega_(omega+1)) in the fast growing hierarchy.
X_X+1 & X_X | ... X_4 & X_3 & X_2 & X & 10 ~ {X,X+2/2} & 10
X_X+1^^X & X_X | ... X_4 & X_3 & X_2 & X & 10
psi0(Omega_(omega+2)) in fgh. you might get the idea by now so let's do some skipping.
{X,X+3/2} & 10
{X,2X/2} & 10
{X,X^2/2} & 10
{X,X^X/2} & 10
{X,X^X^X/2} & 10
{X,X^^X/2} & 10
Bimixommwil = ψ(Ω_ψ(Ω)) in the fast-growing hierarchy
By now its probably more reasonable to put some "Maksudovisms", or googolisms by Denis Maksudov.
{X,X^^^X/2} & 10
{X,{X,X,X}/2} & 10
{X,{X,X(1)2}/2} & 10
{X,{X,X(0,1)2}/2} & 10
{X,X^^X&10/2} & 10
{X,{X,3/2}/2} & 10
{X,{X,4/2}/2} & 10
{X,{X,X/2}/2} & 10
Trimixommwil = ψ(Ω_ψ(Ω_ψ(Ω))) in the fast-growing hierarchy
{X,4,2/2} & 10
Quadrimixommwil = ψ(Ω_ψ(Ω_ψ(Ω_ψ(Ω)))) in the fast-growing hierarchy
{X,5,2/2} & 10
{X,X,2/2} & 10 ~ Binommwil = ψ(Ω_Ω) in the fast-growing hierarchy
we reached there.
Dangerrouse = {10,10/10} && 10
Dangerrouseplex = {10,10/dangerrouse} && 10
{10,10/1,2} && 10
{10,10/10,10} && 10
{10,10/1/2} && 10
Dutriakulus = {3,3//2} = 3 && 3 && 3 = {3,3(/1)2} && 3
Dutetrakulus = {4,4//2} = 4 && 4 && 4 && 4 = {4,4(/1)2} && 4 && 4
Dudekulus = {10,10//2}
Trickulus = {L,2}10,100 = {10,100//2}
Array structure catched up with the actual value again.
Or, the fgh ordinal "catched-up" to the sgh ordinal again.
If you want to know the ordinal itself, it's ψ(Ω_Ω_ω).
Nucleatrixul = 200![[[200 200] 200] 200]
Tridecodutwix = {10,10,10//2}
{X,X,2//2} & 10 ~ Trinommwil = ψ(Ω_Ω_Ω) in FGH
Tridekulus = {10,10///2}
Millennulus = {10,100///2}
Nucleaquaxul = 200![[[[200 200] 200] 200] 200]
{X,X,2///2} & 10 ~ Quadrinommwil = ψ(Ω_Ω_Ω_Ω) in FGH
Tetradekulus = {10,10////2}
Quatreldulus = {10,100////2}
Quinteldulus = {10,100/////2}
...
Grand century = {L,100}10,10 = {10,10///...100 slashes...///2} = {10,100(1)/2}
Big Hoss = {L,100}100,100 = {100,100///...100 slashes...///2} = {100,100(1)/2}
Was Bowers' dog that big.
Here is the limit of Extended Buchholz' OCF.
Since at the time of the writing I'm not versed in FGH ordinals even beyond the SVO, I'm just gonna put whatever Douglas Shamlin jr. put. anyway I was relying heavily on his "analysis".
By the way, the array structure techically catched up to the number itself ω times by now.
The ordinal in the OCF that Douglas Shamlin Jr. uses in his newest 2024 edition of the large number list video,
is psi(psi_i(0)). basically psi(psi_i(z)) enumerates fixed-points of the Ω_n function.
The first one is this one.
We also reached the Limit of Bird's Array Notation.
Grand Hoss = {100,100///...100 slashes...///100}
Great big Hoss = {L,big hoss}big hoss,big hoss = {big hoss,big hoss///...big hoss slashes...///2}
Since BEAF is not really well-defined, I will have to go with FGH & SAN.
in SAN we have reached approximately s(100,100{1,,1{1,,1,,2}2}2).
f_ψ(ψI(0)^ψI(0)) (10)
f_ψ(ψI(0)^ψI(0)^ω) (10)
f_ψ(ψI(0)^ψI(0)^ψI(0)) (10)
f_ψ(Ω_(ψI(0)+1)) (10)
We still have a while to get to {L,X+1}10,10.
f_ψ(Ω_(ψI(0)+ω)) (10)
f_ψ(Ω_Ω_(ψI(0)+1)) (10)
f_ψ(ψI(1)) (10)
Nope... not yet...
f_ψ(ψI(2)) (10)
f_ψ(ψI(3)) (10)
f_ψ(ψI(ω)) (10)
f_ψ(ψI(Ω)) (10)
f_ψ(ψI(Ω_ω)) (10)
f_ψ(ψI(Ω_Ω)) (10)
f_ψ(ψI(Ω_Ω_ω)) (10)
f_ψ(ψI(ψI(0))) (10) ~ s(10, 10 { 1,,1 {1,,1,,2} 1 {1,,1{1,,1,,2}2} 2 } 2) ~ s(X,2X{1,,1{1,,1,,2}2}2) & 10
f_ψ(ψI(ψI(ψI(0)))) (10) ~ s(10, 10 { 1,,1 {1,,1,,2} 1 {1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2} 2 } 2) ~ s(X,3X{1,,1{1,,1,,2}2}2) & 10
I had to space it out.
Unimah = f_ψ(I) (10) ~ s(10,10{1,,1{1,,1,,2}1{1,,1,,2}2}2) ~ s(X,X^2{1,,1{1,,1,,2}2}2) & 10
We reached somewhere here. but our "array structure catching counter" is still at ω.
Why? Because since it hit omega, the array structure diverged from the number itself.
But we will reach ω+1 soon, but we still have a climb... ok fine lets go for it -_-
f_ψ(I^2) (10) ~ s(10,10{1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2}2) ~ s(X,X^3{1,,1{1,,1,,2}2}2) & 10
f_ψ(I^ω) (10) ~ s(10,10{1,,1{2,,1,,2}2}2) ~ s(X,X^X{1,,1{1,,1,,2}2}2) & 10
f_ψ(I^Ω) (10) ~ s(10,10{1,,1{1{1,,2}2,,1,,2}2}2) ~ s(X,X,2{1,,1{1,,1,,2}2}2) & 10
f_ψ(I^Ω_2) (10) ~ s(10,10{1,,1{1{1,,3}2,,1,,2}2}2) ~ s(X,X{1,,1{1{1,,2}2,,1,,2}2}2) & 10
f_ψ(I^Ω_3) (10) ~ s(10,10{1,,1{1{1,,4}2,,1,,2}2}2) ~ s(X,X{1,,1{1{1,,3}2,,1,,2}2}2) & 10
Mirongualith = {L,X+1}10,100 = {10,100/(1)/2} ~ f_ψ(I^Ω_ω) (100) ~ s(10,100{1,,1{1{1,,1,2}2,,1,,2}2}2) ~ s(X,100{1,,1{1{1,,1,2}2,,1,,2}2}2) & 10
Finally the array structure catching counter went up to ω+1. you can see that by looking at the SAN structures.
f_ψ(I^Ω_Ω) (10) ~ s(10,10{1,,1{1{1,,1{1,,2}2}2,,1,,2}2}2) ~ s(X,X,2{1,,1{1{1,,1,2}2,,1,,2}2}2) & 10
Mirongualithplex = {L,X+2}10,100 = {10,100//(1)/2} ~ f_ψ(I^Ω_Ω_ω) (100) ~ s(10,100{1,,1{1{1,,1{1,,1,2}2}2,,1,,2}2}2) ~ s(X,100{1,,1{1{1,,1{1,,1,2}2}2,,1,,2}2}2) & 10
Mirongualithduplex = {L,X+3}10,100 = {10,100///(1)/2} ~ f_ψ(I^Ω_Ω_Ω_ω) (100) ~ s(10,100{1,,1{1{1{1,,1{1,,1,2}2}2}2,,1,,2}2}2) ~ s(X,100{1,,1{1{1,,1{1,,1{1,,1,2}2}2}2,,1,,2}2}2) & 10
Miringualith = {L,2X}10,100 = {L,X+100}10,10 = {10,100(1)//2} ~ f_ψ(I^ψI(0)) (100) ~ s(10,100{1,,1{1{1,,1{1,,1,,2}2}2,,1,,2}2}2)
Mirangualith = {L,3X}10,100 = {L,2X+100}10,10 = {10,100(1)///2} ~ f_ψ(I^ψI(I^ψI(0))) (100) ~ s(10,100{1,,1{1{1,,1{1{1,,1{1,,1,,2}2}2,,1,,2}2}2,,1,,2}2}2)
Mireengualith = {L,4X}10,100 = {L,3X+100}10,10 = {10,100(1)////2} ~ f_ψ(I^ψI(I^ψI(I^ψI(0)))) (100) ~ s(10,100{1,,1{1{1,,1{1{1,,1{1{1,,1{1,,1,,2}2}2,,1,,2}2}2,,1,,2}2}2,,1,,2}2}2)
The sequence is getting long... so let's go to...
Birongualith = {L,X^2}10,100 = {L,100X}10,10 = {10,100(1)(1)/2} ~ f_ψ(I^I) (100) ~ s(10,100{1,,1{1,,1,,2}2,,1,,2}2)
Ok so we may be going a bit slow. If you really are interested in the in-betweens, you can view Douglas Shamlin Jr's large number list video at exactly this point here .
Trirongualith = {L,X^3}10,100 = {10,100(1)(1)(1)/2} ~ f_ψ(I^2I) (100) ~ s(10,100{1,,1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}2}2)
Quadrirongualith = {L,X^4}10,100 = {10,100(1)(1)(1)(1)/2} f_ψ(I^3I) (100) ~ s(10,100{1,,1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}2}2)
Bukuwaha = {L,X^X}100,100 = {100,100(2)/2} ~ f_ψ(I^Iω) (100) ~ s(100,100{1,,1{2{1,,1,,2}2,,1,,2}2}2)
f_ψ(I^IΩ) (10) ~ s(10,10{1,,1{1{1,,2}2{1,,1,,2}2,,1,,2}2}2) ~ s(X,X,2{1,,1{2{1,,1,,2}2,,1,,2}2}2) & 10
f_ψ(I^I(Ω_2)) (10) ~ s(10,10{1,,1{1{1,,3}2{1,,1,,2}2,,1,,2}2}2) ~ s(X,X{1,,1{1{1,,2}2{1,,1,,2}2,,1,,2}2}2) & 10
f_ψ(I^I(Ω_3)) (10) ~ s(10,10{1,,1{1{1,,4}2{1,,1,,2}2,,1,,2}2}2) ~ s(X,X{1,,1{1{1,,3}2{1,,1,,2}2,,1,,2}2}2) & 10
f_ψ(I^I(Ω_4)) (10) ~ s(10,10{1,,1{1{1,,5}2{1,,1,,2}2,,1,,2}2}2) ~ s(X,X{1,,1{1{1,,4}2{1,,1,,2}2,,1,,2}2}2) & 10
f_ψ(I^I(Ω_ω)) (10) ~ s(10,10{1,,1{1{1,,1,2}2{1,,1,,2}2,,1,,2}2}2) ~ s(X,X{1,,1{1{1,,1,2}2{1,,1,,2}2,,1,,2}2}2) & 10 ~ {L,(X^X)+1}10,10
Our catching counter? It's at X^X + 1
By now, the array strucure's catching up really frequently. it's kinda useless to show the structure.
The catching counter for now is the same as the thing after the {L, in the beaf array.
{L,(X^X)+X}10,10 = {10,10(1)/(2)/2} ~ f_ψ(I^I(ψI(0))) (10) ~ s(10,10{1,,1{1{1,,1{1,,1,,2}2}2{1,,1,,2}2,,1,,2}2}2)
{L,(X^X)*2}10,10 = {10,10(2)//2} ~ f_ψ(I^I(ψI(I^Iω))) (10) ~ s(10,10{1,,1{1{1,,1{2{1,,1,,2}2,,1,,2}2}2{1,,1,,2}2,,1,,2}2}2)
{L,X^(X+1)}10,10 = {10,10(2)(1)/2} ~ f_ψ(I^I^2) (10) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}2}2)
{L,X^(X+1)+1}10,10 = {10,10/(2)(1)/2} ~ f_ψ(I^((I^2) + Ω_ω)) (10) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}2,,1,,2}2}2)
{L,X^(X+1)+X}10,10 = {10,10(1)/(2)(1)/2} ~ f_ψ(I^((I^2) + ψI(0))) (10) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1{1,,1,,2}2}2,,1,,2}2}2)
I'm making two parts green so its easier to understand the SAN approximation.
{L,X^(X+1)*2}10,10 = {10,10(2)(1)//2} ~ f_ψ(I^((I^2) + ψI(I^I^2))) (10) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1{1{1,,1,,2}3,,1,,2}2}2,,1,,2}2}2)
{L,X^(X+2)}10,10 = {10,10(2)(1)(1)/2} ~ f_ψ(I^((I^2) + I) (10) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}2,,1,,2}2)
{L,X^(X+3)}10,10 = {10,10(2)(1)(1)(1)/2} ~ f_ψ(I^((I^2) + 2I) (10) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}2}2)
Miplingualith = {L,X^2X}10,100 = {10,100(2)(2)/2} ~ f_ψ(I^((I^2) + Iω) (100) ~ s(10,100{1,,1{1{1,,1,,2}3,,1,,2}1{2{1,,1,,2}2,,1,,2}2}2)
Finally another named number. If you are not interested in seeing all this, just scroll down.
But I would recommend not to!
{L,(X^2X)+1}10,10 = {10,10/(2)(2)/2} ~ f_ψ(I^((I^2) + IΩ_ω) (100) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,2}2{1,,1,,2}2,,1,,2}2}2)
{L,(X^2X+1)}10,10 = {10,10(2)(2)(1)/2} ~ f_ψ(I^((I^2) * 2) (100) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}3,,1,,2}2}2)
{L,(X^2X+2)}10,10 = {10,10(2)(2)(1)(1)/2} ~ f_ψ(I^((I^2) * 2 + I) (100) ~ s(10,10{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}2,,1,,2}2}2)
Maplingualith = {L,X^3X}10,100 = {10,100(2)(2)(2)/2} ~ f_ψ(I^((I^2) * 2 + Iω) (100) ~ s(10,100{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}3,,1,,2}1{2{1,,1,,2}2,,1,,2}2}2)
{L,X^(3X+1)}10,10 = {10,100(2)(2)(2)(1)/2} ~ f_ψ(I^((I^2) * 3) (100) ~ s(10,100{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}3,,1,,2}2}2)
Meeplingualith = {L,X^4X}10,100 = {10,100(2)(2)(2)(2)/2} ~ f_ψ(I^((I^2) * 3 + Iω)) (100) ~ s(10,100{1,,1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}3,,1,,2}1{1{1,,1,,2}3,,1,,2}1{2{1,,1,,2}2,,1,,2}2}2)
You get the idea. Just check back at the SAN structure.
Boplingualith = {L,X^X^2}10,100 = {10,100(3)/2} ~ f_ψ(I^((I^2) * ω)) (100) ~ s(10,100{1,,1{2{1,,1,,2}3,,1,,2}2}2)
Troplingualith = {L,X^X^3}10,100 = {10,100(4)/2} ~ f_ψ(I^((I^3) * ω)) (100) ~ s(10,100{1,,1{2{1,,1,,2}4,,1,,2}2}2)
Quadroplingualith = {L,X^X^4}10,100 = {10,100(5)/2} ~ f_ψ(I^((I^4) * ω)) (100) ~ s(10,100{1,,1{2{1,,1,,2}5,,1,,2}2}2)
Quintoplingualith = {L,X^X^5}10,100 = {10,100(6)/2} ~ f_ψ(I^((I^5) * ω)) (100) ~ s(10,100{1,,1{2{1,,1,,2}6,,1,,2}2}2)
You... get... the... idea.
Moduplingualith = {L,X^X^X}10,100 = {10,100(0,1)/2} ~ f_ψ(I^I^ω) (100) ~ s(10,100{1,,1{2{1,,1,,2}1,2,,1,,2}2}2)
Motriplingualith = {L,X^X^X^X}10,100 = {10,100((1)1)/2} ~ f_ψ(I^I^I^ω) (100) ~ s(10,100{1,,1{1{2,,1,,2}2,,1,,2}2}2)
Goomerdust = {L,X^^X}10,100 ~ f_ψ(Ω_(I+1)) (100) ~ s(10,100{1{1,,2,,2}2,,1,,2}2)
Our old catching counter is at epsilon-zero. {L,<X-structure>} corresponds to ord. structure where ord.structure is the catching counter.
ω^^ω = ε0.
{L,(X^^X)+1}10,10 ~ f_ψ(Ω_(I+1)^(Ω_ω)) (10) ~ s(10,10{1{1{1,,1,2}2,,2,,2}2,,1,,2}2)
{L,(X^^X)*X}10,10 ~ f_ψ(Ω_(I+1)^I) (10) ~ s(10,10{1{1{1,,1,,2}2,,2,,2}2,,1,,2}2)
{L,X^^X | 2}10,10 ~ f_ψ(Ω_(I+1)^I+Ω_(I+1)) (10) ~ s(10,10{1{1,,2,,2}2{1{1,,1,,2}2,,2,,2}2,,1,,2}2)
Remember Terrible Tethrathoth from Part 6? That was about X^^X | 2 & 100 in Bower's X-Structures.
That also means that the catching counter's at ε1.
{L,(X^^X | 2) * X}10,10 ~ f_ψ((Ω_(I+1)^I)2) (10) ~ s(10,10{1{1{1,,1,,2}2,,2,,2}3,,1,,2}2)
{L,X^^(X+1)}10,10 ~ f_ψ((Ω_(I+1)^I)ω) (10) ~ s(10,10{1{1{1,,1,,2}2,,2,,2}1,2,,1,,2}2)
Gimmerdust = {L,X^^2X}10,100
Boomerdust = {L,X^^X^2}10,100
Goomerdarxitri = {L,X^^X^^X}10,100
Cosmiculus = {L,X^^^X}10,100 ~ f_ψ(Ω_(I+1) ^ Ω_(I+1)) (100) ~ s(10,100{1{1{1,,2,,2}2,,2,,2}2,,1,,2}2)
Terribulus = {L,X^^^^X}10,100 ~ f_ψ(Ω_(I+1) ^ (Ω_(I+1) * 2)) (100) ~ s(10,100{1{1{1,,2,,2}2,,2,,2}1{1{1,,2,,2}2,,2,,2}2,,1,,2}2)
{L,{X,X,X}}10,10 ~ f_ψ(Ω_(I+1) ^ (Ω_(I+1) * ω)) (10) ~ s(10,10{1{2{1,,2,,2}2,,2,,2}2,,1,,2}2)
{L,{X,X,1,2}}10,10 ~ f_ψ(Ω_(I+1) ^ (Ω_(I+1) ^ 2)) (10) ~ s(10,10{1{1{1,,2,,2}3,,2,,2}2,,1,,2}2)
{L,{X,X(1)2}}10,10 ~ f_ψ(Ω_(I+1) ^ (Ω_(I+1) ^ ω)) (10) ~ s(10,10{1{1{1,,2,,2}1,2,,2,,2}2,,1,,2}2)
The catching counter reached ψ(Ω^Ω^ω), or the Small Veblen Ordinal!!
{L,{X,X,2(1)2}}10,10 ~ f_ψ(Ω_(I+1) ^ (Ω_(I+1) ^ Ω_(I+1))) (10) ~ s(10,10{1{1{1,,2,,2}1{1,,2,,2}2,,2,,2}2,,1,,2}2)
{L, X_2^^X & X}10,10 ~ f_ψ(Ω_(I+2)) (10) ~ s(10,10{1{1,,3,,2}2,,2,,2}2)
{L, X_2^^^X & X}10,10 ~ f_ψ(Ω_(I+2) ^ Ω_(I+2)) (10) ~ s(10,10{1{1{1,,3,,2}2,,3,,2}2,,2,,2}2)
{L, X_3 ^^ X & X_2 & X}10,10 ~ f_ψ(Ω_(I+3)) (10) ~ s(10,10{1{1,,4,,2}2,,3,,2}2)
{L, X_4 ^^ X & X_3 & X_2 & X}10,10 ~ f_ψ(Ω_(I+4)) (10) ~ s(10,10{1{1,,5,,2}2,,4,,2}2)
First gardener = {L, L}10,100 ~ f_ψ(Ω_(I+ω)) (100) ~ s(10,100{1,,1,2,,2}2)
NOW THE CATCHING COUNTER REACHED LEGION SPACE!
It's at ψ(Ω_ω) in the fast-growing hierarchy.
f_ψ(Ω_(I+ω2)) (10) ~ s(10,10{1,,1,3,,2}2) ~ s(X,2X{1,,1,2,,2}2) & 10
We're gonna "slow down" (in googological sense!) to understand the new FSes.
So we'll need the array structure again.
f_ψ(Ω_(I+ψ(Ω))) (10) ~ s(10,10{1,,1{1`2}2,,2}2) ~ s(X,X^^X{1,,1,2,,2}2) & 10
f_ψ(Ω_(I+Ω)) (10) ~ s(10,10{1,,1{1,,2}2,,2}2) ~ s(X,X,2{1,,1,2,,2}2) & 10
f_ψ(Ω_(I+Ω_2)) (10) ~ s(10,10{1,,1{1,,3}2,,2}2) ~ s(X,X{1,,1{1,,2}2,,2}2) & 10
f_ψ(Ω_(I+Ω_3)) (10) ~ s(10,10{1,,1{1,,4}2,,2}2) ~ s(X,X{1,,1{1,,3}2,,2}2) & 10
f_ψ(Ω_(I+Ω_4)) (10) ~ s(10,10{1,,1{1,,5}2,,2}2) ~ s(X,X{1,,1{1,,4}2,,2}2) & 10
{L, L+1}10,10 ~ f_ψ(Ω_(I+Ω_ω)) (10) ~ s(10,10{1,,1{1,,1,2}2,,2}2) ~ s(X,X{1,,1{1,,1,2}2,,2}2) & 10
Now, finally we bunked up the catching counter to ψ(Ω_ω) + 1.
{L,2L}10,10 ~ f_ψ(Ω_(I+ψI(Ω_(I+ω))) (10) ~ s(10,10{1,,1{1,,1{1,,1,2,,2}2}2,,2}2)
{L,XL}10,10 ~ f_ψ(Ω_(I*2)) (10) ~ s(10,10{1,,1{1,,1,,2}2,,2}2)
Everyone knows that XL is bigger than L. (a fun pun.)
we bunked up the catching counter to ψ((Ω_ω) + 1).
f_ψ(Ω_(I*2+1)) (10) ~ s(10,10{1,,2{1,,1,,2}2,,2}2)
we bunked up the catching counter to ψ(Ω_(ω + 1)).
f_ψ(Ω_(I*2+2)) (10) ~ s(10,10{1,,3{1,,1,,2}2,,2}2)
we bunked up the catching counter to ψ(Ω_(ω + 2)).
f_ψ(Ω_(I*2+ω)) (10) ~ s(10,10{1,,1,2{1,,1,,2}2,,2}2)
we bunked up the catching counter to ψ(Ω_(ω2)).
f_ψ(Ω_(I*3)) (10) ~ s(10,10{1,,1{1,,1,,2}3,,2}2)
we bunked up the catching counter to ψ(Ω_(ω2) + 1).
f_ψ(Ω_(I*ω)) (10) ~ s(10,10{1,,1{1,,1,,2}1,2,,2}2)
f_ψ(Ω_Ω_(I+1)) (10) ~ s(10,10{1,,1{1,,2,,2}2,,2}2)
we bunked up the catching counter to ψ(Ω_Ω).
f_ψ(Ω_Ω_(I+2)) (10) ~ s(10,10{1,,1{1,,3,,2}2,,2}2)
{L,{L,2}}10,10 ~ f_ψ(Ω_Ω_(I+ω)) (10) ~ s(10,10{1,,1{1,,1,2,,2}2,,2}2)
we bunked up the catching counter to ψ(Ω_Ω_ω).
{L,{L,3}}10,10 ~ f_ψ(Ω_Ω_Ω_(I+ω)) (10) ~ s(10,10{1,,1{1,,1{1,,1,2,,2}2,,2}2,,2}2)
{L,{L,X}}10,10 ~ f_ψ(ψI2(0)) (10) ~ s(10,10{1,,1{1,,1,,3}2,,2}2)
And now the catching counter reached ψ(ψI(0)).
Remember the "big" hoss? now the catching counter reached THAT ORDINAL!!
f_ψ(ψI2(ω)) (10) ~ s(10,10{1,,1{1,,1,,3}1,2,,2}2) ~ s(X,X+1{1,,1{1,,1,,3}2,,2}2)
f_ψ(I_2) (10) ~ s(10,10{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2) ~ s(X,X^2{1,,1{1,,1,,3}2,,2}2)
Remember "unimah"? now that's the catching counter!!
{L,{L,X + 1}}10,10 ~ f_ψ(I_2 ^ Ω_I+ω) (10) ~ s(10,10{1,,1{1{1,,1,2,,2}2,,1,,3}2,,2}2)
{L,{L,X + 2}}10,10 ~ f_ψ(I_2 ^ Ω_Ω_I+ω) (10) ~ s(10,10{1,,1{1{1,,1{1,,1,2,,2}2,,2}2,,1,,3}2,,2}2)
{L,{L,2X}}10,10 ~ f_ψ(I_2 ^ ψI2(0)) (10) ~ s(10,10{1,,1{1{1,,1{1,,1,,3}2,,2}2,,1,,3}2,,2}2)
{L,{L,2X + 1}}10,10 ~ f_ψ(I_2 ^ ψI2(I_2 ^ Ω_I+ω)) (10) ~ s(10,10{1,,1{1{1,,1{1{1,,1,2,,2}2,,1,,3}2,,2}2,,1,,3}2,,2}2)
{L,{L,3X}}10,10 ~ f_ψ(I_2 ^ ψI2(I_2 ^ ψI2(0))) (10) ~ s(10,10{1,,1{1{1,,1{1{1,,1{1,,1,,3}2,,2}2,,1,,3}2,,2}2,,1,,3}2,,2}2)
{L,{L,X^2}}10,10 ~ f_ψ(I_2 ^ I_2) (10) ~ s(10,10{1,,1{1{1,,1,,3}2,,1,,3}2,,2}2)
{L,{L,X^3}}10,10 ~ f_ψ(I_2 ^ (I_2 * 2)) (10) ~ s(10,10{1,,1{1{1,,1,,3}2,,1,,3}1{1{1,,1,,3}2,,1,,3}2,,2}2)
{L,{L,X^X}}10,10 ~ f_ψ(I_2 ^ (I_2 * ω)) (10) ~ s(10,10{1,,1{2{1,,1,,3}2,,1,,3}2,,2}2)
{L,{L,X^(X+1)}}10,10 ~ f_ψ(I_2 ^ I_2 ^ 2) (10) ~ s(10,10{1,,1{1{1,,1,,3}3,,1,,3}2,,2}2)
{L,{L,X^(X^2+1)}}10,10 ~ f_ψ(I_2 ^ I_2 ^ 3) (10) ~ s(10,10{1,,1{1{1,,1,,3}4,,1,,3}2,,2}2)
{L,{L,X^(X^X)}}10,10 ~ f_ψ(I_2 ^ I_2 ^ ω) (10) ~ s(10,10{1,,1{1{1,,1,,3}1,2,,1,,3}2,,2}2)
{L,{L,X^(X^X+1)}}10,10 ~ f_ψ(I_2 ^ I_2 ^ I_2) (10) ~ s(10,10{1,,1{1{1,,1,,3}1{1,,1,,3}2,,1,,3}2,,2}2)
{L,{L,X^(X^X^X+1)}}10,10 ~ f_ψ(I_2 ^ I_2 ^ I_2 ^ I_2) (10) ~ s(10,10{1,,1{1{1{1,,1,,3}1{1,,1,,3}2,,1,,3}2,,1,,3}2,,2}2)
{L,{L,X^^X}}10,10 ~ f_ψ(Ω_(I_2 + 1)) (10) ~ s(10,10{1{1,,2,,3}2,,1,,3}2)
{L,{L,X_2 ^^X&X}}10,10 ~ f_ψ(Ω_(I_2 + 2)) (10) ~ s(10,10{1{1,,3,,3}2,,1,,3}2)
{L,{L,X_3 ^^X&X_2&X}}10,10 ~ f_ψ(Ω_(I_2 + 3)) (10) ~ s(10,10{1{1,,4,,3}2,,1,,3}2)
{L,{L,L}}10,10 ~ f_ψ(Ω_(I_2 + ω)) (10) ~ s(10,10{1,,1,2,,3}2)
remember {L,L}? it was at s(10,10{1,,1,2,,2}2). now we "leveled it up" to 1,,1,2,,3.
{L,{L,{L,L}}}10,10 ~ f_ψ(Ω_(I_3 + ω)) (10) ~ s(10,10{1,,1,2,,4}2)
{L,{L,{L,{L,L}}}}10,10 ~ f_ψ(Ω_(I_4 + ω)) (10) ~ s(10,10{1,,1,2,,5}2)
AND SO ON...
{L,X,2}10,10 ~ f_ψ(I_ω) (10) ~ s(10,10{1,,1,,1,2}2)
YESSSSSS! WE FINALLY MADE IT HERE!! THE COUNTER HAS CAUGHT UP WITH THE ORDINAL ITSELF!!!!
We can denote this with a Ω, as the first fixed-point.
f_ψ(I_ω^2) (10) ~ s(10,10{1,,1,,1,1,2}2) ~ s(X,X^2{1,,1,,1,2}2) & 10
f_ψ(I_Ω) (10) ~ s(10,10{1,,1,,1{1,,2}2}2) ~ s(X,X,2{1,,1,,1,2}2) & 10
f_ψ(I_Ω_ω) (10) ~ s(10,10{1,,1,,1{1,,1,2}2}2) ~ s(X,X{1,,1,,1{1,,1,2}2}2) & 10 ~ {L,{L,X,2}+1}10,10
f_ψ(I_ψI(I_ω)) (10) ~ s(10,10{1,,1,,1{1,,1{1,,1,,1,2}2}2}2) ~ {L,{L,X,2}*2}10,10
Uninotos = f_ψ(I_I) (10) ~ s(10,10{1,,1,,1{1,,1,,2}2}2) ~ {L,{L,X,2}*X}10,10
Finally another named number
f_ψ(I_Ω_(I+ω)) (10) ~ s(10,10{1,,1,,1{1,,1,2,,2}2}2) ~ {L,{L,{L,X,2}+1}}10,10
f_ψ(I_Ω_(I_2+ω)) (10) ~ s(10,10{1,,1,,1{1,,1,2,,3}2}2) ~ {L,{L,{L,{L,X,2}+1}}}10,10
f_ψ(I_I_ω) (10) ~ s(10,10{1,,1,,1{1,,1,,1,2}2}2) ~ {L,X,2} | 2 10,10
Catched up again. basically now the catching counter is Ω*2.
Binotos = f_ψ(I_I_I) (10) ~ s(10,10{1,,1,,1{1,,1,,2{1,,1,,2}2}2}2) ~ {L,{L,X,2} | 2 * X}10,10
We can have trinotos, quadrinotos, etc.
Now remember when we were using ψI(0) as the fixed-point for Ω_Ω_Ω_Ω_...?
now we can use ψI(2,0)(0) as a fixed-point for I_I_I_I_...
We can skip all the way to...
{L,X+1,2}10,10 = {L,X,2} | L 10,10 ~ f_ψ(Ω_(I(2,0) + ω)) (10) ~ s(10,10{1,,1,2,,1,,2}2)
{L,X+2,2}10,10 = {L,X,2} | {L,L} 10,10 ~ f_ψ(Ω_(I_(I(2,0)+1) + ω)) (10) ~ s(10,10{1,,1,2,,2,,2}2)
{L,2X,2}10,10 = {L,X,2} | {L,X,2} 10,10 ~ f_ψ(I_(I(2,0)+ω)) (10) ~ s(10,10{1,,1,,1,2,,2}2)
{L,3X,2}10,10 ~ f_ψ(I_(I(2,1)+ω)) (10) ~ s(10,10{1,,1,,1,2,,3}2)
{L,4X,2}10,10 ~ f_ψ(I_(I(2,2)+ω)) (10) ~ s(10,10{1,,1,,1,2,,4}2)
{L,X^2,2}10,10 ~ f_ψ(I(2,ω)) (10) ~ s(10,10{1,,1,,1,,1,2}2)
Our catching counter reached Ω^2 !!
{L,X^3,2}10,10 ~ f_ψ(I(3,ω)) (10) ~ s(10,10{1,,1,,1,,1,,1,2}2)
{L,X^4,2}10,10 ~ f_ψ(I(4,ω)) (10) ~ s(10,10{1,,1,,1,,1,,1,,1,2}2)
{L,X^X,2}10,10 ~ f_ψ(I(ω,0)) (10) ~ s(10,10{1{2··}2}2)
We also passed linear pDAN in SAN!!
Uninimah = f_ψ(I(I(0,0),0)) (10) ~ s(10,10{1{1{1,,2}2··}2}2) ~ s(X,X,2{1{2··}2}2) & 10
Binimah = f_ψ(I(I(I(0,0),0),0)) (10) ~ s(10,10{1{1{1{1{1,,2}2··}2}2··}2}2) ~ s(X,X,2{1{1{1{2··}2}2··}2}2) & 10
{L,X^X,2} | X 10,10 ~ f_ψ(ψI(1,0,0)(0)) (10) ~ s(10,10{1{1,,2··}2}2)
We reached the "most annoying part of SAN". the catching counter's at (Ω^ω) * ω
f_ψ(I(1,0,0)) (10) ~ s(10,10{1{1{1{1,,2··}2}2··}1{1{1{1{1,,2··}2}2··}1,,2}2}2) ~ s(X,X^2{1{1{1{1,,2··}2}2··}2}2) & 10
f_ψ(ψI(1,0,0,0)(0)) (10) ~ s(10,10{1{1{1{1,,2··}2}3··}2}2)
Now we can just treat the I like we used the Veblen Phi Function a long time ago,
and the limit of this "I function" is at...
{L,X^X^X,2}10,10 ~ f_ψ(M^M^ω) (10) ~ s(10,10{1{1{1{1,,2··}2}1,2··}2}2)
And guys, we reached the Mahlo level!
Just like the Ω^Ω^ω was diagonalizing the multiargument phi function, now M^M^ω diagonalizes the I function!!
Tritetremar = f_ψ(M^M^M) (10) ~ {L,X^X^X,2} | X 10,10 ~ s(10,10{1{1{1{1,,2··}2}1{1{1,,2··}2}2··}2}2)
Again, a maksudovism.
The catching counter is at (Ω^ω^ω) * ω.
{L,X^^X,2}10,10 ~ f_ψ(Ω_M+1) (10) ~ s(10,10{1,,2{1,,2··}2}2) = s(10,10{1{1,,2{1,,2··}2}2{1,,2··}2}2)
Catching counter's at Ω^(ε0).
{L,X_2 ^^X&X,2}10,10 ~ f_ψ(Ω_M+2) (10) ~ s(10,10{1,,3{1,,2··}2}2)
{L,L,2}10,10 ~ f_ψ(Ω_M+ω) (10) ~ s(10,10{1,,1,2{1,,2··}2}2)
{L,X,3}10,10 ~ f_ψ(M_ω) (10) ~ s(10,10{1{1,,2··}1,2}2)
Uninemar = f_ψ(M_M) (10) ~ {L,{L,X,3},2} | X 10,10 ~ s(10,10{1{1,,2··}1{1{1,,2··}2}2}2)
Not just another named number. The counter also reached Ω^Ω.
{L,X,4}10,10 ~ f_ψ(M(1,ω)) (10) ~ s(10,10{1{1,,2··}1{1,,2··}1,2}2)
He
{L,2,X}10,10 ~ f_ψ(M(ω,0)) (10) ~ s(10,10{1{2,,2··}2}2)
It's starting to get more and more complicated. I'll just skip to...
Second gardener = {L,X(1)2}10,100 = X @ 10 ~ f_ψ(T^T^ω) (100) ~ s(10,100{1{1,,1,2··}2}2), Catching counter: Ω^Ω^ω
Note I'm using UNOCF.
{L,X,2(1)2}10,10 = X_2 @ 10 ~ f_ψ((T^T^T) * ω) (10) ~ s(10,10{1{1,,1,,2··}1,2}2), Catching counter: Ω^Ω^Ω
Incredible first gardener = {L,L,L,L,... first gardener Ls ...,L,L,L,L(1)X}10,100
{L,X(0,1)2}10,10 = X_2 ^ X @ 10 ~ f_ψ(T^T^T^ω) (10) ~ s(10,10{1{1{2··}2··}2}2), Catching counter: Ω^Ω^Ω^ω
Towporgulus = {L,L((1)1)2}10,100
s(L,X{1`2}2)10,10 = X_2 ^^ X @ 10 ~ f_ψ(Ω_(T+1)) (10) ~ s(10,10{1{1`,,2··}2}2), Catching counter: ε(Ω+1)
We hit the limit of pDAN!
s(L,X,2{1`2}2)10,10 ~ X_2 ^^ X_2 @ 10 ~ s(10,10{1{1{1,,2`··}2··}1,2}2), Catching counter: ε(Ω*2) = ψ_Ω_2(Ω_2 * Ω)
s(L,X{1,,3}2)10,10 ~ X_3 ^^ X & X_2 @ 10 ~ s(10,10{1{1,,`3··}2}2), cc: ψ_Ω_2(Ω_3)
s(L,X{1,,4}2)10,10 ~ X_4 ^^ X & X_3 & X_2 @ 10 ~ s(10,10{1{1,,`4··}2}2), cc: ψ_Ω_2(Ω_4)
Largornity = L @ 10 ~ s(L,X{1,,1,2}2)10,10 ~ s(10,10{1{1,,`1,2··}2}2), cc: ψ_Ω_2(Ω_ω)
Nice largornity = {L,X} @ 10 ~ s(L,X{1,,1,,2}2)10,10 ~ s(10,10{1{1,,`1,,`2··}2}2), cc: ψ_Ω_2(I)
Reversed triakulus = {3,3\2} = 3 @ 3 @ 3 ~ s(3,3{1{1{1,,`2··`}1,2··}2}2), cc: ψ_Ω_2(M_ω)
X_3 @ X_2 @ 10 ~ s(10,10{1{1{1,,`1,2··`}2··}2}2) ~ f_ψ(T_2^T_2^ω) (10), cc: ψ_Ω_2(T^T^ω)
X_4 @ X_2 @ 10
X_4 ^^ X @ X_2 @ 10 ~ s(10,10{1,,,3}2)
{L,1}X_4,X @ X_2 @ 10
{L,X,2}X_4,X @ X_2 @ 10
X_5 @ X_4 @ X_2 @ 10
X_6 @ X_4 @ X_2 @ 10
X_6 ^^ X @ X_4 @ X_2 @ 10 ~ s(10,10{1,,,4}2)
X_8 ^^ X @ X_6 @ X_4 @ X_2 @ 10 ~ s(10,10{1,,,5}2)
Farfarfarmoxxyerpowdyer = {10,100\2} = {L2,1}10,100 ~ s(10,100{1,,,1,2}2) ~ f_ψ(T_ω) (100)
YES AND IT'S OFFICIAL, we reached triple comma in SAN and lugion space in BEAF!!
I guess by now Y-sequence hit Y(1,2,4,8,14), and BMS hit (0,0,0)(1,1,1)(2,2,1)! A MILESTONE!
Farfarfarmoxxyerpowdyer howdreyer = {L,L,100\2}10,100
...
{10,10\\2} = {L2,2}10,10 ~ s(10,10{1,,,1{1,,,1,2}2}2) ~ f_ψ(T_T_ω) (10)
{10,10\\\2} = {L2,3}10,10 ~ s(10,10{1,,,1{1,,,1{1,,,1,2}2}2}2) ~ f_ψ(T_T_ω) (10)
{10,10\\\\2} = {L2,4}10,10 ~ s(10,10{1,,,1{1,,,1{1,,,1,2}2}2}2) ~ f_ψ(T_T_T_ω) (10)
Goshomity = {L2,100}100,100 ~ s(100,100{1,,,1,,,2}2) ~ f_ψ(T(1,0)) (100)
Good goshomity = {L2,goshomity}100,100 ~ s(100,3,2{1,,,1,,,2}2) ~ f_ψ(T(1,0)) (f_ψ(T(1,0)) (100))
Great goshomity = {L2,good goshomity}100,100 ~ s(100,4,2{1,,,1,,,2}2)
Big bukuwaha = {L2,{L,X^X}}100,100
Gishomity = {L2,X,2}100,100
Gashomity = {L2,X,3}100,100
Geeshomity = {L2,X,4}100,100
{L2,X,X}10,10
Moshomity = {L2,10,100}100,100
Boshomity = {L2,10,10,100}100,100
Troshomity = {L2,10,10,10,100}100,100
Goshobity = {L2,X(1)2}100,100
Gishobity = {L2,X,2(1)2}100,100
Goshotrity = {L2,X(1)3}100,100
{L2,X(0,1)2}10,10
s(L2,X{1`2}2)10,10 ~ X_2 ^^ X % 10 ~ s(10,10{1{1,,,2,,,2}2,,,1,,,2}2)
I could have listed more, but let's just jump to...
{L3,1}10,10 = {10,10|2} ~ s(10,10{1,,,1,,,1,2}2) ~ f_ψ(C(2;;ω))) (10)
lagion space??
{L3,X}10,10 ~ s(10,10{1,,,1,,,1,,,2}2) ~ f_ψ(C(2;;1,0))) (10)
{L3,L2}10,10 ~ s(10,10{1,,,1,2,,,1{1,,,1,,,1,,,2}2}2) ~ f_ψ(T_(C(2;;1,0))+ω)) (10)
{L3,{L2,X}}10,10 ~ s(10,10{1,,,1,,,2{1,,,1,,,1,,,2}2}2) ~ f_ψ(T(1,C(2;;1,0)+1)) (10)
Bongo bukuwaha = {L3,{L2,{L,X^X}}}100,100 ~ s(100,100{1,,,2,,,2{1,,,1,,,1,,,2}2}2) ~ ψ(I_T(1,C(2;;1,0)+1)+1) (100)
{L3,L3}10,10 ~ s(10,10{1,,,1,,,1,2{1,,,1,,,1,,,2}2}2) ~ f_ψ(C(2;;C(2;;1,0)+ω)) (10)
{L3,X,2}10,10 ~ s(10,10{1,,,1,,,1{1,,,1,,,1,,,2}1,2}2) ~ f_ψ(C(2;;1,ω)) (10)
{L3,X,3}10,10 ~ s(10,10{1,,,1,,,1{1{1,,,1,,,1,,,2}2,,,1,,,1,,,2}1,2}2) ~ f_ψ(C(2;;1;ω)) (10)
{L4,1}10,10 = {10,10-2} ~ s(10,10{1,,,1,,,1,,,1,2}2) ~
ligion space
Quabinga bukuwaha = {L4,{L3,{L2,{L,X^X}}}}100,100 ~ s(100,100{1,,,2,,,2,,,2{1,,,1,,,1,,,1,,,2}2}2)
{L5,1}10,10 ~ s(10,10{1,,,1,,,1,,,1,,,1,2}2)
"lowgion" space
Hazukashi bukuwaha = {L5,{L4,{L3,{L2,{L,X^X}}}}}100,100 ~ s(100,100{1,,,2,,,2,,,2,,,2{1,,,1,,,1,,,1,,,1,,,2}2}2)
{L6,1}10,10 ~ s(10,10{1,,,1,,,1,,,1,,,1,,,1,2}2)
"luggion" space
Queen bukuwaha = {L6,{L5,{L4,{L3,{L2,{L,X^X}}}}}}100,100 ~ s(100,100{1,,,2,,,2,,,2,,,2,,,2{1,,,1,,,1,,,1,,,1,,,1,,,2}2}2)
{L7,1}10,10 ~ s(10,10{1,,,1,,,1,,,1,,,1,,,1,,,1,2}2)
"laigion" space
Meameamealokkapoowa = {L100,10}10,10 ~ s(10,100{1{2···}2}2)
Another original bowerism.
{LL,1}10,10
Meameamealokkabipoowa = {LL100,10}10,10
{LLL,1}10,10
Meameamealokkapoowa oompa = {LLL...{L100,10}10,10 array of L's...LLL,10}10,10
Meameamealokkapoowa oompa loompa = {LLL...meameamealokkapoowa oompa array of L's...LLL,10}10,10
LIMIT OF BEAF ~ s(10,10{1{1,,,2···}1,2}2)
s(10,10{1,,,,1,2}2) ~ (0,0,0)(1,1,1)(2,2,1)(2,2,1)
s(10,10{1,,,,,1,2}2) ~ (0,0,0)(1,1,1)(2,2,1)(2,2,1)(2,2,1)
s(10,10{1,,,,,,1,2}2) ~ (0,0,0)(1,1,1)(2,2,1)(2,2,1)(2,2,1)(2,2,1)
(0,0,0)(1,1,1)(2,2,1)(3,0,0) ~ f_ψ(C(1{ω}0)) (10) Catching counter: Ω_ω
Yes!! We reached the Small Dropping Ordinal (SDO) which is ψ(C(1{ω}0)) and (0,0,0)(1,1,1)(2,2,1)(3,0,0) in bms.
SAN number = S_64 ~ f_ψ(C(1{ω}0))+1 (64)
S_0 = 10.
S_n = s(10,10{1,,,...S_n-1 commas...,,,1,2}2)
This number is S_64.
BMS: (0,0,0)(1,1,1)(2,2,1)(3,0,0)(0,0,0)
LIMIT OF SAN
(0,0,0)(1,1,1)(2,2,1)(3,0,0)(1,0,0)
(0,0,0)(1,1,1)(2,2,1)(3,0,0)(1,1,0)
...
Now I will just show the FGH ordinal and catching counter next to BMS.
Also, I'm using UNOCF.
(0,0,0)(1,1,1)(2,2,1)(3,0,0)(2,2,1) ~ ψ(C(1{ω+1}0))
(0,0,0)(1,1,1)(2,2,1)(3,0,0)(2,2,1)(3,0,0) ~ ψ(C(1{ω*2}0))
(0,0,0)(1,1,1)(2,2,1)(3,0,0)(3,0,0) ~ ψ(C(1{ω^2}0))
(0,0,0)(1,1,1)(2,2,1)(3,0,0)(4,0,0) ~ ψ(C(1{ω^ω}0))
(0,0,0)(1,1,1)(2,2,1)(3,0,0)(4,1,0) ~ ψ(C(1{ε0}0))
(0,0,0)(1,1,1)(2,2,1)(3,1,0) ~ ψ(C(1{Ω}0))
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(1,1,1) ~ ψ(C(1{Ω_ω}0)) Catching counter: Ω_ω + 1
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(1,1,1)(2,2,1) ~ ψ(C(1{T_ω}0)) Catching counter: Ω_ω + Ω_2
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(1,1,1)(2,2,1)(3,0,0) ~ ψ(C(1{C(1{ω}0)}0)) Catching counter: Ω_ω * 2
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,0,0) ~ ψ(C(1{1,0}0)) Catching counter: Ω_ω * ω
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,1,1)(3,1,1) ~ ψ(C(1{1,0}ω)) Catching counter: Ω_ω * Ω
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,1,1)(3,2,1) ~ ψ(C(1{1,0}1;;ω)) Catching counter: Ω_ω * Ω_2
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,1,1)(3,2,1)(4,0,0) ~ ψ(C(1{1,0}1{ω}0)) Catching counter: Ω_ω ^ 2
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,1,1)(3,2,1)(4,1,0)(2,0,0) ~ ψ(C(2{1,0}0)) Catching counter: Ω_ω ^ 2 * ω
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,1,1)(3,2,1)(4,1,0)(3,0,0) ~ ψ(C(ω{1,0}0)) Catching counter: Ω_ω ^ ω
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,0) ~ ψ(Ω_C(1{1,1}0)+1) Catching counter: ε_((Ω_ω)+1)
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1) ~ ψ(C(1{1,1}ω)) Catching counter: Ω_(ω+1)
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1)(2,2,1) ~ ψ(C(1{1,2}ω)) Catching counter: Ω_(ω+2)
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1)(2,2,1)(2,2,1) ~ ψ(C(1{1,3}ω)) Catching counter: Ω_(ω+3)
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1)(3,0,0) ~ ψ(C(1{1,ω}0)) Catching counter: Ω_(ω*2)
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1)(3,1,0)(2,0,0) ~ ψ(C(1{2,0}0)) Catching counter: Ω_(ω*2) * ω
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1)(3,1,0)(2,2,1)(3,1,0)(2,0,0) ~ ψ(C(1{3,0}0)) Catching counter: Ω_(ω*3) * ω
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(3,0,0) ~ ψ(C(1{ω,0}0)) Catching counter: Ω_(ω^2)
(0,0,0)(1,1,1)(2,2,1)(3,1,0)(3,1,0)(2,0,0) ~ ψ(C(1{1,0,0}0)) Catching counter: Ω_(ω^2) * ω
...
(0,0,0)(1,1,1)(2,2,1)(3,1,1) ~ ψ(C(1{1;0}ω)) Catching counter: Ω_Ω
(0,0,0)(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,0,0) ~ ψ(C(1{1{ω}0}0)) Catching counter: Ω_Ω_ω
(0,0,0)(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1) ~ ψ(C(1{1{1;0}0}ω)) Catching counter: Ω_Ω_Ω
(0,0,0)(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)(6,2,1)(7,0,0) ~ ψ(C(1{1{1{ω}0}0}0)) Catching counter: Ω_Ω_Ω_ω
(0,0,0)(1,1,1)(2,2,1)(3,2,0) ~ ψ(C(1:0)) Catching counter: ψ_I(I)
We have gone so far that UNOCF has been obliterated!! I'll be using Strong exUNOCF now.
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(2,2,1)(3,2,0) ~ ψ(C(2:0)) Catching counter: ψ_I(I*2)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(3,0,0) ~ ψ(C(ω:0)) Catching counter: ψ_I(I*ω)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(3,1,1) ~ ψ(C(1;0:ω)) Catching counter: ψ_I(I*Ω)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(3,2,0) ~ ψ(C(1:0:0)) Catching counter: ψ_I(I^2)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(3,2,0)(3,2,0) ~ ψ(C(1:0:0:0)) Catching counter: ψ_I(I^3)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(4,0,0) ~ ψ(C(1:;0)) Catching counter: ψ_I(I^ω)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(4,1,1) ~ ψ(C(1:;0)^C(1:;0)) Catching counter: ψ_I(I^Ω)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(4,2,0) ~ ψ(C(1:;0)^C(1:;0)^C(1:;0)) Catching counter: ψ_I(I^I)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(4,3,0) ~ ψ(Ω_C(1:;0)+1) Catching counter: ψ_I(Ω_I+1)
(0,0,0)(1,1,1)(2,2,1)(3,2,0)(4,3,0)(5,4,1)(6,4,0) ~ ψ(ψI(C(1:;0))) Catching counter: ψ_I(Ω_I+1)
(0,0,0)(1,1,1)(2,2,1)(3,2,1) ~ ψ(C(1:;ω)) Catching counter: C(1,0) = I
The catching counter reached inaccessibles. hey, all of these numbers are inaccessible! they're too big
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(3,0,0) ~ ψ(C(ω:;0)) Catching counter: I_ω
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(3,2,1) ~ ψ(C(1:;0:;ω)) Catching counter: I(2,0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(3,2,1)(3,2,1) ~ ψ(C(1:;0:;0:;ω)) Catching counter: I(3,0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,0,0) ~ ψ(C(1:;;0)^C(1:;;0)^ω) Catching counter: I(ω,0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,0)(3,2,1) Catching counter: I(1,0,0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1) ~ ψ((C(1:;;0)^C(1:;;0)^C(1:;;0))*ω) Catching counter: C(1;0) = M
Wow the catching counter reached a mahlo!
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(3,2,1)(4,2,1) ~ ψ((C(1:;;0)^C(1:;;0)^(C(1:;;0)*2))*ω) Catching counter: C(2;0) = M(...,0,0,0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(3,2,1)(4,2,1)(3,2,1)(4,2,1) ~ ψ((C(1:;;0)^C(1:;;0)^(C(1:;;0)*3))*ω) Catching counter: C(3;0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(4,0,0) ~ ψ(C(1:;;0)^C(1:;;0)^(C(1:;;0)*ω)) Catching counter: C(ω;0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(4,2,0)(3,2,1) ~ ψ(C(1:;;0)^C(1:;;0)^C(1:;;0)^2) Catching counter: C(1,0;0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(4,2,1) ~ ψ((C(1:;;0)^C(1:;;0)^C(1:;;0)^2)*ω)) Catching counter: C(1;0;0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(4,2,1)(4,2,1) ~ ψ((C(1:;;0)^C(1:;;0)^C(1:;;0)^3)*ω)) Catching counter: C(1;0;0;0)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(5,0,0) ~ ψ(C(1:;;0)^C(1:;;0)^C(1:;;0)^ω) Catching counter: ψT(T^T^ω)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(5,2,1) ~ ψ((C(1:;;0)^C(1:;;0)^C(1:;;0)^C(1:;;0))*ω) Catching counter: ψT(T^T^T)
(0,0,0)(1,1,1)(2,2,1)(3,2,1)(4,2,1)(5,2,1)(6,2,1) ~ ψ((C(1:;;0)^^5)*ω) Catching counter: ψT(T^^4)
(0,0,0)(1,1,1)(2,2,1)(3,3,0) ~ ψ(Ω_(C(1:;;0)+1)) Catching counter: ψT(Ω_(T+1))
This ordinal is called the "second back gear ordinal" or SBGO short.
(0,0,0)(1,1,1)(2,2,1)(3,3,1) ~ ψ(C(1:;;ω)) Catching counter: C(1;;0) = T
Now we need T's for the Catching counter!!
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,0,0) ~ ψ(C(ω:;;0)) Catching counter: T_ω
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1) ~ ψ(C(1:;0:;;ω)) Catching counter: C(1;;1,0) = T(1,0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(3,2,1) ~ ψ(C(1:;1:;0:;;ω)) Catching counter: C(1;;2,0) = T(2,0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,0,0) ~ ψ(C(1:;;0:;;0)^C(1:;;0:;;0)^ω) Catching counter: C(1;;ω,0) = T(ω,0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,2,0)(3,2,1) ~ ψ(C(1:;;0:;;0)^C(1:;;0:;;0)^C(1:;;0:;;0)) Catching counter: C(1;;1,0,0) = T(1,0,0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,2,1) ~ ψ((C(1:;;0:;;0)^C(1:;;0:;;0)^C(1:;;0:;;0))*ω) Catching counter: C(1;;1;0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1) ~ ψ(C(1:;;0:;;ω)) Catching counter: C(2;;0) = ψX(X^2)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1) ~ ψ(C(2:;;0:;;ω)) Catching counter: C(2;;1) = ψX(X^2*2)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(3,0,0) ~ ψ(C(ω:;;0:;;0)) Catching counter: C(2;;ω) = ψX(X^2*ω)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(3,2,1)(4,3,1) ~ ψ(C(1:;;0:;;0:;;ω)) Catching counter: C(3;;0) = ψX(X^3)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(4,0,0) ~ ψ(C(1:;;;0)^C(1:;;;0)^ω) Catching counter: C(ω;;0) = ψX(X^ω)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(4,2,1)(5,3,1) ~ ψ((C(1:;;;0)^C(1:;;;0)^C(1:;;;0))*ω) Catching counter: C(1;;0;;0) = ψX(X^X)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(4,2,1)(5,3,1)(4,2,1)(5,3,1) ~ ψ((C(1:;;;0)^C(1:;;;0)^C(1:;;;0^2))*ω) Catching counter: C(1;;0;;0;;0) = ψX(X^X^2)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(4,2,1)(5,0,0) ~ ψ(C(1:;;;0)^C(1:;;;0)^C(1:;;;0)^ω) Catching counter: ψX(X^X^ω)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,2,1)(4,3,1)(4,2,1)(5,3,1)(5,2,1)(6,3,1) ~ ψ((C(1:;;;0)^C(1:;;;0)^C(1:;;;0)^C(1:;;;0)) * ω) Catching counter: ψX(X^X^X)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,3,0) ~ ψ(Ω_(C(1:;;;0)+1)) Catching counter: ψX(Ω_(X+1))
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,3,1) ~ ψ(C(1:;;;ω)) Catching counter: C(1;;;0) = X
Wow now X's??
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(3,3,1)(3,3,1) ~ ψ(C(1:;;;;ω)) Catching counter: C(1;;;;0) = Y
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,0,0) ~ ψ(C(1:{ω}0)) Catching counter: C(1{ω}0)
Now let's speed up to...
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,3,1) ~ ψ(C(1::;ω)) Catching counter: C(1:0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,3,1)(3,3,1)(4,3,1) ~ ψ(C(2::;ω)) Catching counter: C(2:0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,3,1)(4,3,1) ~ ψ(C(1::;0::;ω)) Catching counter: C(1:0:0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,4,1)(5,0,0) ~ ψ(C(1::{ω}0)) Catching counter: C(1:{ω}0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,4,1)(5,5,1)(6,0,0) ~ ψ(C(1:::{ω}0)) Catching counter: C(1::{ω}0)
(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,4,1)(5,5,1)(6,6,1)(7,0,0) ~ ψ(C(1::::{ω}0)) Catching counter: C(1:::{ω}0)
(0,0,0)(1,1,1)(2,2,2) ~ ψ(C(1{:ω}0)) Catching counter: C(1{:ω}0) ~ Y(1,2,4,8,15)
WE FINALLY REACHED THE ULTIMATE MILESTONE!! THE CATCHING COUNTER IS THE SAME AS THE FGH CARDINAL WITHOUT THE ψ!
By the way, this ordinal is called the omega back ordinal.
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,0)(2,2,2) ~ ψ(C(1{:1{:ω}0}0)) Catching counter: C(1{:1{:ω}0}0)
Let's make a Tier 2 catching counter that finds out how many times the catching counter catches with the fgh cardinal.
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,0)(3,0,0) ~ ψ(C(1 :_2 0)) Tier-2 cc: ω
There are so many catching counter catching points that we can say that the catching counter has been obliterated. We can now use our new Tier-2 catching counter.
The fgh ordinals will be a bit messy as Strong exUNOCF itself gets obliterated by QSS level, so I'll not show the FGH ordinal anymore.
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,0)(3,2,2)(4,2,0)(2,2,2) Tier-2 cc: ω+1
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,0)(3,2,2)(4,2,0)(2,2,2)(3,2,2)(4,2,0)(2,2,2) Tier-2 cc: ω+2
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,0)(3,2,2)(4,2,0)(2,2,2)(3,2,2)(4,2,0)(3,0,0) Tier-2 cc: ω*2
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,0)(3,2,2)(4,2,0)(3,0,0) Tier-2 cc: ω^2
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,0)(5,3,0) Tier-2 cc: ε0
...
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,1) Tier-2 cc: Ω
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2) ~ ψ(C(1{:_2 ω}0)) Tier-2 cc: C(1{:_2 ω}0) Tier-3 cc: 1
It looks like the catching functions are very weak but that is because BMS is so strong.
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(3,2,2)(4,2,2) ~ ψ(C(1{:_3 ω}0)) Tier-4 cc: 1
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(3,2,2)(4,2,2)(3,2,2)(4,2,2) ~ ψ(C(1{:_4 ω}0)) Tier-5 cc: 1
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(3,2,2)(4,2,2)(3,2,2)(4,2,2)(3,2,2)(4,2,2) ~ ψ(C(1{:_5 ω}0)) Tier-6 cc: 1
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,0,0) ~ ψ(C(1{:_ω 0}0)) Catching counter tier: ω
Now the catching function is COMPLETELY OBLITERATED!!
Strong exUNOCF expressions are a bit messy so let's skip them.
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,1,0)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,1,1)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,2,0)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,2,1)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,2,2)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,0,0)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,1,0)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,1,1)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,2,0)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,2,1)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,2,2)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,2,2)(6,2,2)
(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(5,2,2)(6,2,2)(7,2,2)
(0,0,0)(1,1,1)(2,2,2)(3,3,0)
(0,0,0)(1,1,1)(2,2,2)(3,3,1)
(0,0,0)(1,1,1)(2,2,2)(3,3,2)
(0,0,0)(1,1,1)(2,2,2)(3,3,2)(4,4,2)
(0,0,0)(1,1,1)(2,2,2)(3,3,2)(4,4,2)(5,5,2)
(0,0,0)(1,1,1)(2,2,2)(3,3,3) ~ Y(1,2,4,8,15,26)
NEW MILESTONE!!
(0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,4,4) ~ Y(1,2,4,8,15,26,42)
(0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,4,4)(5,5,5) ~ Y(1,2,4,8,15,26,42,64)
Yukito's sequences can have many patterns. 1,2,4,8,15,26,42,64,... are known as cake numbers.
(0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,4,4)(5,5,5)(6,6,6) ~ Y(1,2,4,8,15,26,42,64,93)
(0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,4,4)(5,5,5)(6,6,6)(7,7,7) ~ Y(1,2,4,8,15,26,42,64,93,130)
(0,0,0,0)(1,1,1,1) ~ Y(1,2,4,8,16) ~ LIMIT OF TSS
Let's finish up the numbers. If you are interested in "Transfinite TSS" then you can take a detour to QSS
(0,0,0,0,0)(1,1,1,1,1) ~ Y(1,2,4,8,16,32) ~ LIMIT OF QSS
(0,0,0,0,0,0)(1,1,1,1,1,1) ~ Y(1,2,4,8,16,32,64) ~ LIMIT OF QiSS
(0,0,0,...,0,0,0)(1,1,1,...,1,1,1) ~ Y(1,3) = YY(1,2,5) ~ LIMIT OF BMS
Y(1,4) ~ LIMIT OF BILINEAR BMS
Y(1,5) ~ LIMIT OF TRILINEAR BMS
Y(1,ω) ~ LIMIT OF PLANAR BMS & Y-SEQUENCE
LIMIT OF REALMIC BMS & YY-SEQUENCE
LIMIT OF FLUNAR BMS
LIMIT OF DIMENSIONAL BMS & ω-Y SEQUENCE
???
Tritar = Tar(3)
Quadritar = Tar(4)
Quintitar = Tar(5)
Sextitar = Tar(6)
Septitar = Tar(7)
Octitar = Tar(8)
Nonitar = Tar(9)
Dekotar = Tar(10)
Hektotar = Tar(100)
Kilotar = Tar(1000)
Quettotar = Tar(10^30)
Unintar = Tar(Tar(10))
Bintar = Tar(Tar(Tar(10)))
Trintar = Tar^4(10)
Quadrintar = Tar^5(10)
Quintintar = Tar^6(10)
Sextintar = Tar^7(10)
Septintar = Tar^8(10)
Octintar = Tar^9(10)
Nonintar = Tar^10(10)
Dekintar = Tar^11(10)
Hektintar = Tar^101(10)
Kilintar = Tar^1001(10)
Quettintar = Tar^(10^30)+1(10)
Tarintar = Tar^(Tar(10)+1)(10)
Loader's number = D(D(D(D(D(99)))))
This entry won in the Bignum Bakeoff contest.
Σ(636)
Σ(643)
Σ(745)
Σ(748)
Σ(1919)
Ξ(1000000)
Σ((2^65536)-1)
Rayo(7339)
Rayo's number = Rayo(10^100)
Large Number Garden Number = f^10(10{10}10)
BIG FOOT = FOOT^10(10^100)
It's ill-defined, and so's everything below... We don't know the precise order...
Little Bigeddon
Oblivion
Utter Oblivion
Sasquatch
Sam's Number
And... the final entry...
Infinity (∞)