Seeing as how many people are having difficulties keeping track of the various googological realms, I've decided to to create my own googological ruler which aims to be as straightforward and as easy to use as possible.
I call my googological levels "Psi Levels", and they are simply integers between 0 to 1000. The proposed numbers are the bottom boundary of each level.
Please tell me if you find this useful. Feedback and suggestions are welcome.
| Proposed Psi Level | Arrows/BEAF | Equivalent Ordinal | Letter Notation | Wiki Class Name |
|---|---|---|---|---|
| 0 | 0 | Class 0 | ||
| 1 | 10 | E1 | Class 1 | |
| 2 | 10↑10 | 1 | E10 | Class 2 |
| 3 | 10↑↑3 | F3 | Class 3 | |
| 4 | 10↑↑4 | F4 | Class 4 | |
| 5 | 10↑↑5 | F5 | Class 5 | |
| 6 | 10↑↑10 | 2 | F10 = G2 | Tetration Level |
| 7 | 10↑↑↑3 | G3 | Up-arrow Notation Level | |
| 8 | 10↑↑↑10 | 3 | G10 | |
| 9 | 10↑↑↑↑10 | 4 | H10 = J4 | |
| 10 | 10↑↑↑↑↑↑↑↑↑↑10 | 10 | J10 = K2 | |
| 11 | {10,10,1010} | ω | K2-1-2 | Linear Omega Level |
| 12 | {10,3,1,2} | K3 | ||
| 13 | {10,10,1,2} | ω+1 | K10 = L10 | |
| 14 | {10,3,2,2} | L3 | ||
| 15 | {10,10,2,2} | ω+2 | L10 = M2 | |
| 16 | {10,10,3,2} | ω+3 | M3 | |
| 17 | {10,10,10,2} | ω×2 | M10 | |
| 18 | {10,10,1,3} | ω×2+1 | N2.1 | |
| 19 | {10,10,10,3} | ω×3 | N3 | |
| 20 | {10,10,10,10} | ω×10 | N10 = P2 | |
| 21 | {10,10,10,1010} | ω2 | P2-100-... | Quadradic Omega Level |
| 22 | {10,10,1,1,2} | ω2+1 | P2-101 | |
| 23 | {10,10,10,1,2} | ω2+ω | P2-11 | |
| 24 | {10,10,10,10,2} | ω2×2 | P2-2 | |
| 25 | {10,10,10,10,3} | ω2×3 | P2-3 | |
| 26 | {10,10,10,10,10} | ω3 | P3 | Polynomial Omega Level |
| 27 |
{10,10,10,10,10,2} |
ω3×2 | P3-2 | |
| 28 | 6 & 10 | ω4 | P4 | |
| 29 | 7 & 10 | ω5 | P5 | |
| 30 | 12 & 10 | ω10 | P10 | |
| 31 | {10,1010 (1) 2) | ωω | Q2-1-10-100-... | Exponentiated Linear Omega Level |
| 32 | {10,10,2 (1) 2) | ωω+1 | Q2-1-10-101 | |
| 33 | {10,10,10 (1) 2) | ωω+ω | Q2-1-10-11 | |
| 34 | {10,10 (1) 3) | ωω×2 | Q2-1-10-2 | |
| 35 | {10,10 (1) 10) | ωω+1 | Q2-1-11 | |
| 36 | {10,10 (1) 10,10) | ωω+2 | Q2-1-12 | |
| 37 | {10,10 (1)(1) 2) | ωω×2 | Q2-1-2 | |
| 38 | {10,10 (2) 2) | ωω2 | Q2-2 | Exponentiated Polynomial Omega Level |
| 39 | {10,10 (3) 2) | ωω3 | Q2-3 | |
| 40 | {10,10 (0,1) 2) | ωω10 | Q3 | Double Exponentiated Polynomial Omega Level |
| 41 | {10,1010 (0,1) 2) | ωωω | Q3-1-10-... | |
| 42 | {10,10 (1,1) 2} | ωωω+1 | Q3-1-11 | |
| 43 | {10,10 (0,2) 2} | ωωω×2 | Q3-1-2 | |
| 44 | {10,10 (0,0,1) 2} | ωωω2 | Q3-2 | |
| 45 | {10,10 ((1)) 2} | ωωωω | Q4 | Triple Exponentiated Polynomial Omega Level |
| 46 | {10,10 ((2)) 2} | ωωωω2 | Q4-2 | |
| 47 | {10,10 ((0,1)) 2} | ω↑↑5 | Q5 | Iterated Cantor Normal Form Level |
| 48 | {10,10 (((1))) 2} | ω↑↑6 | Q6 | |
| 49 | {10,10 ((((1)))) 2} | ω↑↑8 | Q8 | |
| 50 | ω↑↑10 | Q10 | ||
| 51 | ε0 | R2.0-1-100-10-10-100-... | Epsilon Level | |
| 52 | ε0+1 | R2.0-1-100-10-10-101 | ||
| 53 | ε0+ω | R2.0-1-100-10-10-11 | ||
| 54 | ε0×2 | R2.0-1-100-10-10-2 | ||
| 55 | ε0×ω | R2.0-1-100-10-11 | ||
| 56 | ε02 | R2.0-1-100-10-2 | ||
| 57 | ε0ω | R2.0-1-100-11 | ||
| 58 | ε0↑↑2 | R2.0-1-100-2 | ||
| 59 | ε0↑↑3 | R2.0-1-100-3 | ||
| 60 | ε0↑↑10 | R2.0-1-101 | ||
| 61 | ε1+1 | R2.0-1-101-100-100-1001 | ||
| 62 | ε1×2 | R2.0-1-101-100-100-2 | ||
| 63 | ε12 | R2.0-1-101-100-2 | ||
| 64 | ε1↑↑2 | R2.0-1-101-2 | ||
| 65 | ε2 | R2.0-1-102 | ||
| 66 | ε3 | R2.0-1-103 | ||
| 67 | εω | R2.0-1-11 | ||
| 68 | εε0 | R2.0-1-2 | ||
| 69 | εεε0 | R2.0-1-3 | ||
| 70 | ζ0 | R2.0-2 | Binary Phi Level | |
| 71 | ζ0×2 | R2.0-2-100000-100-100-2 | ||
| 72 | ζ02 | R2.0-2-100000-100-2 | ||
| 73 | ζ0↑↑2 | R2.0-2-100000-2 | ||
| 74 | εζ0+1 | R2.0-2-100001 | ||
| 75 | ζ1 | R2.0-2-10001 | ||
| 76 | ζ2 | R2.0-2-1002 | ||
| 77 | ζω | R2.0-2-101 | ||
| 78 | ζζ0 | R2.0-2-2 | ||
| 79 | ζζζ0 | R2.0-2-3 | ||
| 80 | η0 | R2.0-3 | ||
| 81 | η1 | R2.0-3-10001 | ||
| 82 | ηω | R2.0-3-1001 | ||
| 83 | ηη0 | R2.0-3-2 | ||
| 84 | φ(4,0) | R2.0-4 | ||
| 85 | φ(5,0) | R2.0-5 | ||
| 86 | φ(6,0) | R2.0-6 | ||
| 87 | φ(7,0) | R2.0-7 | ||
| 88 | φ(8,0) | R2.0-8 | ||
| 89 | φ(10,0) | R2.1 | ||
| 90 | φ(ω,0) | R2.1-... | ||
| 91 | φ(ω,1) | R2.1-1-10-1001 | ||
| 92 | φ(ω+1,0) | R2.1-1-11 | ||
| 93 | φ(ω×2,0) | R2.1-1-2 | ||
| 94 | φ(ω2,0) | R2.1-2 | ||
| 95 | φ(ωω,0) | R2.2 | ||
| 96 | φ(ε0,0) | R3 | ||
| 97 | φ(φ(ω,0),0) | R3.1 | ||
| 98 | φ(φ(ε0,0),0) | R4 | ||
| 99 | φ(φ(φ(ε0,0),0),0) | R5 | ||
| 100 | Γ0 = ψ(ΩΩ) = φ(1,0,0) | R10 = S2 | Bachmann's Collapsing Level | |
| 110 | φ(1,0,0,0) = ψ(ΩΩ2) | S3 | ||
| 120 | SVO = ψ(ΩΩω) | S10 = T2 | ||
| 130 | LVO = ψ(ΩΩΩ) | T10 = V2 | ||
| 140 | ψ(ΩΩΩΩ) | V3 | ||
| 150 | BHO = ψ(ψ2(0)) | V10 = W2 | Higher Computable Level | |
| 160 | ψ(ψ3(0)) | W3 | ||
| 170 | ψ(Ωω) | W10 | ||
| 180 | ψ(ΩΩ) | |||
| 190 | ψ(ΩΩΩ) | |||
| 200 | ψ(ψɪ(0)) | |||
| 210 | ψ(ψɪ(1)) | |||
| 220 | ψ(εɪ+1) = PTO(KPI) | |||
| 230 | ψ(ψɪ(1,0)(0)) | |||
| 240 | ψ(εM+1) | |||
| 250 | fPTO(KP+Π3-r)(F10) | ψ(εK+1) = PTO(KP+Π₃-ref) | ||
| 260 | fPTO(KP+Πn-r)(F10) | PTO(KP+Πn-ref) | ||
| 270 | collapse of smallest doubly-stable ordinal [?] | |||
| 280 | collapse of stable ordinal under a nonproj. ordinal [?] | |||
| 290 | fPTO(Π12−CA)(F10) | PTO (Π12−CA) | ||
| 300 | fZ2(10↑↑10) | PTO (2nd order arithmetic) | ||
| 310? | fZ3(10↑↑10) | PTO (3rd order arithmetic) | ||
| 320? | D(10↑↑10) | Loader's Ordinal | ||
| 400? | fZFC(10↑↑10) | PTO of ZFC | ||
| Reserved for larger computable constructs yet to be discovered | ||||
| 500 | BB(10↑↑10) | ω1ck | (Busy Beavers) | Uncomputables |
| 510 | BB2(10↑↑10) | ω1ck+1 | (Iterated BB) | |
| 520 | BBO(10↑↑10) | ω1ck2 | (Busy Beavers w/ Simple Halting Oracle) | |
| 530 | ω2ck | |||
| 540 | ω3ck | |||
| 550 | ωωck | |||
| 580 | ωω1ck | |||
| 600 |
α = ωαck |
|||
| 700 | Rayo(1010) | |||
| <710 | Fish Number 7 | |||
| 1000 | ω | ω1 | (unattainable) | |