List of notations

Finite Notations

BP

Bx.Py = x^y

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

Enhanced BP

Bx.Py = x^y

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} = a^b

{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} = a^bc

{a, b, c, 0, d} = a↑^dbc

{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}

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

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

{a, b, c_x} = {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)↑(x^y), 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, ω^ω] = α^α↑2+1+1

...

NOBAN 1.0

[a, b] = ab

[a, ω] = a^a

[a, ω+b] = a^b

[a, ω·2] = a↑↑a

[a, ω·b] = a↑↑b-2

[a, ω²] = a↑↑↑a

[a, ω^b] = a↑↑↑b-2

[a, ω^ω] = a↑↑↑a^a-1

[a, ω↑ω^ω] = a↑↑↑↑a

...

NOBAN 2.0

[a, b] = ab

[a, ω] = a^a

[a, ω+b] = a↑(a·a^b)

[a, ω·2] = a↑(a·a↑↑a)

[a, ω·b] = a↑(a·a↑^ba)

[a, ω²] = a↑(a·a↑^(a↑a-1)a)

[a, ω^b] = a↑(a·a↑^{(a↑(a↑...(a-1...))) with b-1 a's}a)

[a, ω^ω] = a↑(a·a↑^{(a↑(a↑...(a-1...))) with a↑a-1 a's}a)

...

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) = ω^α

τ_α+1(β) = τ_α^β(β)

τ_α(β, γ) = τ_α^γ+1(β)

...

Lambda

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

λ(α, β) = τ_α(β)

λ(α, β, γ) = λ(α, λ(α, β, γ-1)) ≠ τ_α(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) = ε_n-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) = ε_ω , κ(1, 1) = ε_ε0, etc.
κ(ω) = ζ₀
For 0 < n < ω, κ(ω + n) = ε_{(ζ0) + n}
For 0 < n < ω, κ(ωn) = ζ_n_{- 1}
For 0 < n < ω, κ(ωⁿ) = φ(1+n, 0)
κ(ω^ω) = Γ₀
κ(ω^ωn) = Γ_n-1
κ(ω^ω↑2) = φ(1, 0, 0, 0)
κ(ω^ω, 0) = Small Veblen ordinal (?)
κ(ω^ω↑ω) = Large Veblen ordinal
κ(ε_ω+1) = Bachmann-Howard ordinal

Rho

C_i⁰(α) = ω_i∪ {0, 1, ω, ω₁} ∪ ρ_i+1(0)

C_i^β+1(α) = {δ + θ, δθ, δ^θ | δ, θ ∈ C_i^β(α)}

C_i(α) = ⋃ C_i^β(α) ∀β < ω

C(α) = ⋃ C_i(α) ∀i < ω

ρ_i(α) = min({γ | γ ∉ C_i(α)})

ρ(α) = min({γ | γ ∉ C(α)})

Xi

∀μ, ν ∈ C_n(α) ⇒ μ + ν ∈ C_n(α)

C₀(α) = α ∪ {0, 1, ω, ω₁} ∪ ω_α

C_n+1(α) = {γ + δ, Ξ_ι(κ) | γ, δ, ι, κ ∈ C_n(α); ι < α} ∪ ω_{n + 1}

Ξ_n(α) = min({λ < ω_α| C_n(α) ∩ ω_α ⊆ α ∧ α ∈ C_n(α)} ∪ ω_α)

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

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

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

For example, the "small" Veblen ordinal = λ₀(Ω^Ω, Ω^Ω↑2), Bachmann-Howard ordinal = λ_Ω(Ω^Ω, Ω^Ω↑Ω), Buchholz's ordinal = λ_Ω(Ω₂, Ω₃), Takeuti-Feferman-Buchholz ordinal = λ_ε(Ω+1)(ε_(Ω2+1), ε_(Ω₃₊₁₎), Extended Buchholz's ordinal = λ_Ω(Ω_Ω, Ω_ΩΩ), ...

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