My Googological Ruler
I've made a googological ruler, a tool to compare big (both computable and uncomputable) numbers.
Inspired by P進大好きbot's Googological Ruler.
Ruler
Here's the googological ruler. This is a scale to compare computable and uncomputable numbers. It assigns each number a range based on its size.
Legend:
Level: number assigned to the range.
Range start: The smallest number in the range.
Range end: The greatest number in the range.
Countable
Level -2: imaginary numbers / fictional goooglogy numbers
Level -1: undefined numbers / ill-defined numbers / infinity / infinite ordinals / infinite cardinals
Level 0: negative numbers / non-integers (this ruler only allows integers)
Level 0.5: 0 ~ 1
Level 1: 2 ~ 6
Level 2: 7 ~ 1,000,000
Level 3: 1,000,001 ~ 10^100
Level 4: 10^100 + 1 ~ 10^10^100
Level 5: 10^10^100 + 1 ~ 10^10^10^100
Level 6: 10^10^10^100 + 1 ~ 10^10^10^10^100
Level 7: 10^10^10^10^100 + 1 ~ f3(10^100)
Level 8: f3(10^100) + 1 ~ f4(10^100)
Level 9: f4(10^100) + 1 ~ fω(10^100)
Level 10: fω(10^100) + 1 ~ fω + 2(10^100)
Level 11: fω + 2(10^100) + 1 ~ fω2(10^100)
Level 12: fω2(10^100) + 1 ~ fω^2(10^100)
Level 13: fω^2(10^100) + 1 ~ fω^ω(10^100)
Level 14: fω^ω(10^100) + 1 ~ fω^ω^ω(10^100)
Level 15: fω^ω^ω(10^100) + 1 ~ fε0(10^100)
Level 16: fε0(10^100) + 1 ~ fζ0(10^100)
Level 17: fζ0(10^100) + 1 ~ fη0(10^100)
Level 18: fη0(10^100) + 1 ~ fφ(ω, 0)(10^100)
Level 19: fφ(ω, 0)(10^100) + 1 ~ fΓ0(10^100)
Level 20: fΓ0(10^100) + 1 ~ fφ(1, 0, 0, 0)(10^100)
Level 21: fφ(1, 0, 0, 0)(10^100) + 1 ~ fψ0(Ω^Ω^ω)(10^100)
Level 22: fψ0(Ω^Ω^ω)(10^100) + 1 ~ fψ0(Ω^Ω^Ω)(10^100)
Level 23: fψ0(Ω^Ω^Ω)(10^100) + 1 ~ fψ0(Ω_2)(10^100)
Level 24: fψ0(Ω_2)(10^100) + 1 ~ fψ0(Ω_3)(10^100)
Level 25: fψ0(Ω_3)(10^100) + 1 ~ fψ0(Ω_ω)(10^100)
Level 26: fψ0(Ω_ω)(10^100) + 1 ~ fψ0(ε(Ω_ω + 1))(10^100)
Level 27: fψ0(ε(Ω_ω + 1))(10^100) + 1 ~ fψ0(Ω_Ω)(10^100)
Level 28: fψ0(Ω_Ω)(10^100) + 1 ~ fψ0(Φ1(0))(10^100)
Level 29: fψ0(Φ1(0))(10^100) + 1 ~ fψΩ1(ψI(0))(10^100)
Level 30: fψΩ1(ψI(0))(10^100) + 1 ~ fψΩ1(I)(10^100)
Level 31: fψΩ1(I)(10^100) + 1 ~ fψΩ1(Λ0)(10^100)
Level 32: fψΩ1(Λ0)(10^100) + 1 ~ fψ0(0, ψ$2(0, 0))(10^100)
Level 33: fψ0(0, ψ$2(0, 0))(10^100) + 1 ~ fψ{χ0(0)}(ψ{χ3(0)}(0))(10^100)
Level 34: fψ{χ0(0)}(ψ{χ3(0)}(0))(10^100) + 1 ~ fψ{χ0(0)}(ψ{χ(M + 1)(0)}(0))(10^100)
Level 35: fψ{χ0(0)}(ψ{χ(M + 1)(0)}(0))(10^100) + 1 ~ fψΩ1(ψ{χ(ε(M + 1))(0)}(0))(10^100)
Level 36: fψΩ1(ψ{χ(ε(M + 1))(0)}(0))(10^100) + 1 ~ fψΩ1(ε(K + 1))(10^100)
Level 37: fψΩ1(ε(K + 1))(10^100) + 1 ~ f{ot(T(Ξ), <)}(10^100)
Level 38: f{ot(T(Ξ), <)}(10^100) + 1 ~ f{ot(T(Υ), <)}(10^100)
Level 39: f{ot(T(Υ), <)}(10^100) + 1 ~ TransInt(2^1,000) (the least transcendental integer)
Level 40: TransInt(2^1,000) + 1 ~ f{PTO(ZFC)}(10^100)
Level 41: f{PTO(ZFC)}(10^100) + 1 ~ all computable numbers
Uncomputable I
Level 0: ill-defined numbers
Level 1: 0 ~ "Proof(n, ⊥)" and all computable numbers
Level 2: "Proof(n, ⊥)" + 1 to Σ(10^100)
Level 3: Σ(10^100) + 1 ~ Σ2(10^100)
Level 4: Σ2(10^100) + 1 ~ Σ10^100(10^100)
Level 5: Σ10^100(10^100) + 1 ~ fω1^CK(10^100)
Level 6: fω1^CK(10^100) + 1 ~ fω_ω^CK(10^100)
Level 7: fω_ω^CK(10^100) + 1 ~ Σ∞(10^100)
Level 8: Σ∞(10^100) + 1 ~ "ℵn = ℵ"
Level 9: "ℵn = ℵ" + 1 ~ Rayo{Σ1}(10^100)
Level 10: Rayo{Σ1}(10^100) + 1 ~ Rayo{Σ100}(10^100)
Uncomputable II
Level 0: ill-defined numbers
Level 1: 0 ~ Rayo's number
Level 2: Rayo's number + 1 ~ Fish number 7
Level 3: Fish number 7 + 1 ~ Rayo{MK}(10^100)
Level 4: Rayo{MK}(10^100) + 1 Rayo{ZFC + I0}(10^100)
Conventions
Ordinal notations and OCFs
The ordinal notation at levels 9 ~ 15 (computable) is Cantor normal form.
The ordinal notation at levels 18 ~ 19, 20 ~ 21 is Veblen function.
The ordinal notation at levels 21 ~ 27 is Buchholz's function.
The ordinal notation at levels 27 ~ 29 is extended Buchholz's function.
The ordinal notation at levels 29 ~ 32, 33 ~ 36 is Rathjen's standard OCF based on a mahlo cardinal.
The ordinal notation at levels 32 ~ 33 is KumaKuma ψ function.
The ordinal notation at levels 36 ~ 37 is Rathjen's standard OCF based on a weakly compact cardinal.
The ordinal notation χ at levels 33 ~ 36 is Rathjen's ordial function.
The Λ0 at levels 31 ~ 32 is the singular cardinal χ_χ_χ ... (0)(0)(0) = ψ{χM(0)}(0).
The ordinal notations (T(Ξ), <) and (T(Y), <) at levels 37 ~ 39 are Stegert's ordinal notations.
P進大好きbot's functions
Transcendental integer system (explained by P進大好きbot in his blog)
Least proof of contradiction (explained by P進大好きbot in his blog)
Busy beaver function (explained by P進大好きbot in his blog)
Aleph (explained by P進大好きbot in his blog)
Rayo's function (explained by P進大好きbot in his blog)
Examples
Numbers in computable
Meameamealokkapoowa oompa: Level -1
フラン数第四形態改二 (= 6): Level 1
Googol: Level 3
Tritri: Level 7
Graham's number: Level 10
Fish number 1: Level 13
Fish number 2: Level 13
Threegold: Level 13
Fish number 3: Level 14
Worm(100): Level 15
Primitive sequence number: Level 16
Fish number 5: Level 16
フラン数第四形態改三: Level 16
Fish number 6: Level 17
TREE(3): unknown (21 ~ 23?)
SCG(13): unknown (25 ~ 27?)
Pair sequence number: Level 26
Laver table: unknown (0 or 10 ~ ?)
Bashicu matrix number: unknown (0 or 36 ~ 39?)
Loader's number: unknown (27 ~ 39?)
Tarintar: unknown (0 or 30 ~ 39?)
The least transcendental integer: Level 40
Numbers in uncomputable I
Σ(10^100): Level 2
Ξ(10^6): unknown (4 ~ 5?)
Fish number 4: Level 5
Numbers in uncomputable II
Croutonillion: Level 0
Sam's number: Level 0
BIG FOOT: Level 0
Little bigeddon: Level 0
Sasquatch: Level 0
Utter oblivion: Level 0
Rayo's number: Level 1
Fish number 7: Level 2
Large number garden number: Level >4