List of notations

Finite Notations

BP

Bx.Py = xy

Bx.Py# with z #s = 10↑↑z topped with Bx.Py#

Enhanced BP

Bx.Py = xy

Bx.Py.Wz = Bx.Py# with z #'s

Bx.Py.Wz# with a #s = Bx.Py# with (10↑↑a topped with z) #'s

BIP-Array

{x, y, z} = 10↑↑y+1 topped with z
{a, b, c, ...d, e, f, ...g, h, i, ...} = {a, {{rest of array}, b, 1}, c}

Going from 2D to 3D is the same as 1D to 2D, etc.

Hyper-Robin

f{0}(n) = 1

f{1}(n) = n

f{2}(n) = 2n

f{n}(n) = n↑2

f{n+1}(n) = n↑3, not (n+1)(n+1)

f{n+2}(n) = n↑4

f{2n}(n) = n↑(n+2)

f{3n}(n) = n↑(2n+2)

...

f{x, y}(n) = (2)(f{x}(n↑y))↑f{x}(n↑y) −1 n

f{x, y, z}(n) = f{{x, y}, {y, z}}(f{n, z}(n))

f{x, y, z, a}(n) = f{{x, y, z}, {y, z, a}}(f{n, x, z}(f{{x, y}, {y, z}, {z, a]}}(n))

f{x, y} (n) = f{f {z, a}}(f{x, y, z, a}(x↑(y↑(z↑a))))

{z, a} {x, y}

Robin-Array

{a, b, ..., n, 0} = {a, b, ... n}

{a, b, ..., n, 1} = {a, b, ... n}

{0, a, b, ...} = 0

{a} = a

{a, b} = ab

{a, b, c} = a↑c b

{a, b, c, d} = {a, b - 1, {a, b, c}, d-1}

{a, b, c, d, e} = {a - 1, b, {a, b, c}, d}↑e {a - 1, b, {a, b, c}, d}


{a, 0, b} = ab

{a, b, 0, c} = abc

{a, b, c, 0, d} = a↑d bc

{a, b, c, d, 0, e} = {a, bc - 1, {a, bc}, d-1}

{a, {b, 0}, c} = {a, b, c} (that’s intuitive, I only put it so you remember)

{a, 0, 1, c} = {a, c, 0, a}

{a, 0, b, c} = {a, b-1, 0, a} (slightly self-contradictory, but whatever)

{a, 0, 0, b, c, d} = {a, 0, d, 1, c, a}


{ax, b, c} = {ax, b, c}

{a, bx, c} = {a, bx, c}

{a, b, cx} = {a, b, cx}

{a{x, y}, b, c} = {ax, ay, b, c}

{a, b{x, y}, c} = {a, bx, by, c}

{a, b, c{x, y}} = {a, b, cx, cy}

{a{x, y, z}, b, c} = {ax, ay, az, b, c}

{a, b{x, y, z}, c} = {a, bx, by, bz, c}

{a, b, c{x, y, z}} = {a, b, cx, cy, cz}


{{x, y, z}a, b, c} = {{xa, ya, za}, {xa, ya, za}, ... {xa, ya, za}} with xyzabc {xa, ya, za}s

{a, {x, y, z}b, c} = {{xb, yb, zb}, {xb, yb, zb}, ... {xb, yb, zb}} with xyzabc {xb, yb, zb}s

...

{a, b, c, #} = {a, b, c, c, c, ... c} with a c’s

{a, b, c, #2} = {a, b, c, c, c, ... c, #} with a c’s

{a, b, c, #3} = {a, b, c, c, c, ... c, #2} with a c’s

...

{a, b, c, #, #} = {a, b, c, #a}

{a, b, c, #, #2} = {a, b, c, #a, #}

{a, b, c, #, #, #} = {a, b, c, #, #a}

...

{a, b, c, #^e} = {a, b, c #, #, #, ... #} with e #s

This allows you to make something like:

{a, b, c, d, #^{e, f, g}}


{a, b, c

d, e, f

g, h, i} = {a, b, c, #^{d, e, f,

g, h, i}} = {a, b, c, #^{d, e, f, #^{g, h, i}}}


Same applies for 3D so you get something like:


{a, b, c, #^{d, e, f, #^{g, h, i, #^{j, k, l, #^{...}}}}} with 8 #^ as opposed to 2, 4D has 26 #^s, etc.


{a, b, c, #^#} = {a, b, c, #^, #}

{a, b, c, *} = {a, b, c, #^#}

{a, b, c, *2} = {a, b, c, #^#, *}

{a, b, c, *3} = {a, b, c, #^#, *2}

...

{a, b, c, **} = {a, b, c, #^#, *a-1}

...

{a, b, c, /d} = {a, b, c, ***...*} with d *s

{a, b, c, /{d, e, f}}

{a, b, c, /{d, e, f, /g}}

{a, b, c, /{d, e, f, /{g, h, i}}}

...

{a, b, c, /#} = {a, b, c, /, /, /, ... /} with a /s

{a, b, c, /##} = {a, b, c, /#a}

...

{a, b, c, /#^d} = {a, b, c, /##...#e} with a #s

...

{a, b, c, /{d, e, f, /g}}

...

{a, b, c, /de} = {a, b, c, /(/(..(./e))...)} with d /s

{a, b, c, //d} = {a, b, c, /ad}

...

{a, b, c, //de} = {a, b, c, //(//...//(e))...)} with d //s

{a, b, c, /^de} = {a, b, c, //.../d} with d /s

...

{a, b, c, 0, \d} = {a, b, c, /ad}

{a, b, c, 0, \de} = {a, b, c, \ \ \...}

{a, b, c, 1, \d} = {a, b, c, 0, \\d}

{a, b, c, n, \d}= {a, b, c, 0, \^n+1 d}

Cataclysm

0[0]0 = 1[0]0 = 0[1]0 = 0[0]1 = 1

a[1]0 = 10↑a

a[b]0 = 10a[b-1]0

a[b, c]0 = 10↑(a↑cb)

a[b, c, d]0 = 10↑(a[b, c-1, d-1]0)

a[b, c, d, e]0 = 10↑(a[b, c-1, d-1, e-1]0)

a[b]c = a[b, b-1]c-1, e.g. 2[2]2 = 2[2, 1]1 = 2[2, 1, 2, 0]0 = 10↑(2[1, 1, 1, -1]0)

a[b]c#d = a[a[a[...b...]]]c with d a's

Super-BP

Bx.Py.Wz#a = (10↑↑(10↑↑a-1)↑z)↑(xy), I think


Bx.Py.Wz#a#b = Bx.Py.Wz#(Bx.Py.Wz#a#b-1)


Bx.Py.Wz##a = Bx.Py.Wz#a#a#a#a#a...#a with a repetitions of #a


Bx.Py.Wz#^#a = Bx.Py.Wz#####...#a with a repetitions of #


Bx.Py.Wz#^^#a = Bx.Py.Wz#^#^#^...^#a with a repetitions of #^


Bx.Py.Wz#^^^#a = Bx.Py.Wz((..((#^^#)^^#)..)^^#)^^#a with a repetitions of ^^#


Bx.Py.Wz#αb = Bx.P(Bx.Py.Wz#α(b-1)).Wz


Bx.Py.Wz#βa = Bx.Py.Wz#α(Bx.Py.Wz#β(b-1))

...

Bx.Py.Wz#Ωa = Bx.Py.Wz#ω(Bx.Py.Wz#Ω(b-1))


NOTE: The greek letter progression goes: lowercase alpha, lowercase beta, lowercase gamma, capital gamma, lowercase delta, capital delta, lowercase epsilon, lowercase zeta, lowercase eta, lowercase theta, capital theta, lowercase lambda, lowercase xi, capital xi, capital phi, capital psi, lowercase omega, capital omega.

OAN

[α] = α

[α, n] = αn

[α, ω] =α2 + 1

[α, ωn] =αn+1 + 1

[α, ωω] = αα↑2+1+1

...

NOBAN 1.0

[a, b] = ab

[a, ω] = aa

[a, ω+b] = ab

[a, ω·2] = a↑↑a

[a, ω·b] = a↑↑b-2

[a, ω2] = a↑↑↑a

[a, ωb] = a↑↑↑b-2

[a, ωω] = a↑↑↑aa-1

[a, ω↑ωω] = a↑↑↑↑a

...

NOBAN 2.0

[a, b] = ab

[a, ω] = aa

[a, ω+b] = a↑(a·ab)

[a, ω·2] = a↑(a·a↑↑a)

[a, ω·b] = a↑(a·a↑ba)

[a, ω2] = a↑(a·a↑(a↑a-1)a)

[a, ωb] = a↑(a·a↑(a↑(a↑...(a-1...))) with b-1 a'sa)

[a, ωω] = a↑(a·a↑(a↑(a↑...(a-1...))) with a↑a-1 a'sa)

...

MU5

{a, b}0 = ab

{a, 1}{c, d} = {a, 1}c = a

{a, b}c = {a, {a, b-1}c}c-1

{a, b}{c, d} = {a, {a, b-1}{c, d}}{c, d-1}

{a, b}{0, d} = {a, a}{b, d-1}

{a, b}{c, 0} = {a, b}c

{a, 0}b = 1

{a, b}{0, 0, c, d} = {a, a}{0, b, c-1, d} = {a, a}{a, b-1, c-2, d-1}

{a, b}{c#1, d} = {a, b}{c, c, c, ..., c#1, d-1}, with b c's

{a, b}{c#n, d} = {a, b}{c#n-1, c#n-1, c#n-1, ..., c#n-1, d-1) with b c#n-1's

{a, b}{c#[0, d], e} = {a, b}, {c#[0, d], e-1#[0,d], e-1#[0, d], ... e-1#[0, d]} with d [0, d]'s

{a, b}{c#[d, e], f} = {a, b}, {c#[d, e], f-1#[d, e], f-1#[d, e], ... f-1#[d, e]} with {d, e}d-1 [d, e]s

{a, b}{c, c, c, ... c#d, 0} # {a, b}{c, c, c, ..., c}

...

Infinite Notations (Ordinal Functions / OCFs)

Tau

τ0(α) = ω+α

τ1(α) = ω·α

τ2(α) = ωα

τα+1(β) = ταβ(β)

τα(β, γ) = ταγ+1(β)

...

Lambda

λ(α, β, γ, ... δ, 1) = λ(α, β, γ, ... δ)

λ(α, β) = τα(β)

λ(α, β, γ) = λ(α, λ(α, β, γ-1)) ≠ τα(β, γ)

λ(α, β, γ, δ) = λ(α, β, λ(α, β, γ, δ-1))

...

Kappa

κ(0) = ω
κ(0, n) = ωn
For 0 < n < ω, κ(n) = εn-1
For n > 0, κ(n, m) = κ(n) but with κ(m) instead of 0 (yeah, it's hard to put into words), e.g. κ(1, 0) = εω , κ(1, 1) = εε0, etc.
κ(ω) = ζ0
For 0 < n < ω, κ(ω + n) = ε(ζ0) + n
For 0 < n < ω, κ(ωn) = ζn - 1
For 0 < n < ω, κ(ωn) = φ(1+n, 0)
κ(ωω) = Γ0
κ(ωωn) = Γn-1
κ(ωω↑2) = φ(1, 0, 0, 0)
κ(ωω, 0) = Small Veblen ordinal (?)
κ(ωω↑ω) = Large Veblen ordinal
κ(εω+1) = Bachmann-Howard ordinal

Rho

Ci0(α) = ωi ∪ {0, 1, ω, ω1} ∪ ρi+1(0)

Ciβ+1(α) = {δ + θ, δθ, δθ | δ, θ ∈ Ciβ(α)}

Ci(α) = ⋃ Ciβ(α) ∀β < ω

C(α) = ⋃ Ci(α) ∀i < ω

ρi(α) = min({γ | γ ∉ Ci(α)})

ρ(α) = min({γ | γ ∉ C(α)})

Xi

∀μ, ν ∈ Cn(α) ⇒ μ + ν ∈ Cn(α)

C0(α) = α ∪ {0, 1, ω, ω1} ∪ ωα

Cn+1(α) = {γ + δ, Ξι(κ) | γ, δ, ι, κ ∈ Cn(α); ι < α} ∪ ωn + 1

Ξn(α) = min({λ < ωα | Cn(α) ∩ ωα ⊆ α ∧ α ∈ Cn(α)} ∪ ωα)

Lambda (again)

NOTE: This OCF I invented is not connected to my Lambda ordinal function, and it is a coincidence that they share the same letter.

λ(α) = ψ(α) in Buchholz's ψ function

λα(β, γ) = limit of the sequence ψ(α), ψ(β), ψ(γ), ...

λαn(β, γ) = ψ(n)th term of the sequence ψ(α), ψ(β), ψ(γ), ...


For example, the "small" Veblen ordinal = λ0(ΩΩ, ΩΩ↑2), Bachmann-Howard ordinal = λΩ(ΩΩ, ΩΩ↑Ω), Buchholz's ordinal = λΩ(Ω2, Ω3), Takeuti-Feferman-Buchholz ordinal = λε(Ω+1)(ε(Ω2+1), ε(Ω3+1)), Extended Buchholz's ordinal = λΩ(ΩΩ, ΩΩΩ), ...