User blog:Rgetar/Designation for the next element of a class above an ordinal and some rules for booster-base

I made up designation for the least element of a class of ordinals C(•) larger than an ordinal α:
 * C(•) ↰ α

Here bullet "•" is placeholder for a parameter, selecting an item from the class C(•). Examples:
 * ε• ↰ α is the next ε number above α
 * φ(1, 0, •) ↰ α = Γ• ↰ α is the next Γ number above α
 * Ω• ↰ α is the next Ω number above α

Some rules for booster-base
Here are some useful rules for the BB (booster-base) system, used in the Ordinal Explorer Online (the full ruleset is here).

Note: the "original" BB has only 3 symbols ("[", "]", "c"), but it can be converted into other formats with more symbols for better readability.

There are 3 kinds of strings in BB: empty string, "c" and [X]β, where X and β are strings. There is also "special" string "Limit" for the least countable ordinal, which cannot be expressed using BB. Formally, "Limit" is not a BB expression.

In [X]β X is called "booster", and β is called "base". For example, in
 * [X][Y][Z]β

booster is X, and base is [Y][Z]β.

So, the rules.


 * 1. Empty string = 0


 * 2. c is some large regular cardinal


 * 3. []β = [0]β = β + 1


 * 4. [[]X]β = [X + 1]β = ...[X][X][X][X][X]β (ω times)


 * 5. If X < Ω or card(X) ≤ β, then [X]β = β + ωX


 * 6. If cof(X) = ω or ω ≤ cof(X) ≤ β, then [X]β = sup{[X[n]]β}, where X[n] are elements of fundamental sequence of X.

(These rules are not enough for all cases).

Some patterns
Here are some patterns using C(•) ↰ α designations. All these patterns are for α < c.

Next uncountable Ω

 * Ω• ↰ α = [c]α (here Ω• are started from Ω, that is ω does not count).

Veblen function
Let Ωx = Ω• ↰ α = [c]α. Then
 * ε• ↰ α = [Ωx]α
 * ζ• ↰ α = [Ωx2]α
 * η• ↰ α = [Ωx3]α
 * φ(4, •) ↰ α = [Ωx4]α
 * φ(5, •) ↰ α = [Ωx5]α
 * φ(ω, •) ↰ α = [Ωxω]α
 * φ(ω + 1, •) ↰ α = [Ωx(ω + 1)]α
 * φ(ε0, •) ↰ α = [Ωxε0]α
 * Γ• ↰ α = [Ωx2]α
 * φ(1, 1, •) ↰ α = [Ωx2 + Ωx]α
 * φ(1, 2, •) ↰ α = [Ωx2 + Ωx2]α
 * φ(1, ω, •) ↰ α = [Ωx2 + Ωxω]α
 * φ(2, 0, •) ↰ α = [Ωx22]α
 * φ(1, 0, 0, •) ↰ α = [Ωx3]α
 * φ(1, 0, 0, 0, •) ↰ α = [Ωx4]α
 * φ(1, 0, 0, 0, 0, •) ↰ α = [Ωx5]α
 * φ(1, 2, 3, 4, 5, •) ↰ α = [Ωx5 + Ωx42 + Ωx33 + Ωx24 + Ωx5]α

Next L
L• are some large regular cardinals (ineffable?), but not so large as c.
 * L• ↰ α = [c2]α

I• (weakly inaccessibles)
Let Lx = L• ↰ α = [c2]α. Then
 * I• ↰ α = [c + Lx]α
 * I(2, •) ↰ α = [c + Lx2]α
 * I(3, •) ↰ α = [c + Lx3]α
 * I(4, •) ↰ α = [c + Lx4]α
 * I(5, •) ↰ α = [c + Lx5]α
 * I(1, 0, •) ↰ α = [c + Lx2]α
 * I(1, 1, •) ↰ α = [c + Lx2 + Lx]α
 * I(1, 2, •) ↰ α = [c + Lx2 + Lx2]α
 * I(2, 0, •) ↰ α = [c + Lx22]α
 * I(1, 0, 0, •) ↰ α = [c + Lx3]α
 * I(1, 0, 0, 0, •) ↰ α = [c + Lx4]α
 * I(1, 0, 0, 0, 0, •) ↰ α = [c + Lx5]α
 * I(1, 2, 3, 4, 5, •) ↰ α = [c + Lx5 + Lx42 + Lx33 + Lx24 + Lx5]α

Φ (Ω fixed points)
Let Ix = I• ↰ α. Then
 * Φ(1, •) ↰ α = [c + Ix]α
 * Φ(2, •) ↰ α = [c + Ix2]α
 * Φ(3, •) ↰ α = [c + Ix3]α
 * Φ(4, •) ↰ α = [c + Ix4]α
 * Φ(5, •) ↰ α = [c + Ix5]α
 * Φ(1, 0, •) ↰ α = [c + Ix2]α
 * Φ(1, 1, •) ↰ α = [c + Ix2 + Ix]α
 * Φ(1, 2, •) ↰ α = [c + Ix2 + Ix2]α
 * Φ(2, 0, •) ↰ α = [c + Ix22]α
 * Φ(1, 0, 0, •) ↰ α = [c + Ix3]α
 * Φ(1, 0, 0, 0, •) ↰ α = [c + Ix4]α
 * Φ(1, 0, 0, 0, 0, •) ↰ α = [c + Ix5]α
 * Φ(1, 2, 3, 4, 5, •) ↰ α = [c + Ix5 + Ix42 + Ix33 + Ix24 + Ix5]α

I•-Φ (I fixed points)
(Designations of this kind were inspired by Scorcher007's list).

Let y = I(2, •) ↰ α. Then
 * I•-Φ(1, •) ↰ α = [c + Lx + y]α
 * I•-Φ(2, •) ↰ α = [c + Lx + y2]α
 * I•-Φ(3, •) ↰ α = [c + Lx + y3]α
 * I•-Φ(4, •) ↰ α = [c + Lx + y4]α
 * I•-Φ(5, •) ↰ α = [c + Lx + y5]α
 * I•-Φ(1, 0, •) ↰ α = [c + Lx + y2]α
 * I•-Φ(1, 2, 3, 4, 5, •) ↰ α = [c + Lx + y5 + y42 + y33 + y24 + y5]α

I(2, •)-Φ (I(2, •) fixed points)
Let y = I(3, •) ↰ α. Then
 * I(2, •)-Φ(1, •) ↰ α = [c + Lx2 + y]α
 * I(2, •)-Φ(2, •) ↰ α = [c + Lx2 + y2]α
 * I(2, •)-Φ(3, •) ↰ α = [c + Lx2 + y3]α
 * I(2, •)-Φ(4, •) ↰ α = [c + Lx2 + y4]α
 * I(2, •)-Φ(5, •) ↰ α = [c + Lx2 + y5]α
 * I(2, •)-Φ(1, 0, •) ↰ α = [c + Lx2 + y2]α
 * I(2, •)-Φ(1, 2, 3, 4, 5, •) ↰ α = [c + Lx2 + y5 + y42 + y33 + y24 + y5]α

I(...)-Φ (I(...) fixed points)

 * I(3, •)-Φ(1, •) ↰ α = [c + Lx3 + I(4, •) ↰ α]α
 * I(4, •)-Φ(1, •) ↰ α = [c + Lx4 + I(5, •) ↰ α]α
 * I(5, •)-Φ(1, •) ↰ α = [c + Lx5 + I(6, •) ↰ α]α
 * I(•, 0)-Φ(1, •) ↰ α = [c + Lx·(I(1, 0, •) ↰ α)]α
 * I(1, 0, •)-Φ(1, •) ↰ α = [c + Lx2 + I(1, 1, •) ↰ α]α
 * I(1, 1, •)-Φ(1, •) ↰ α = [c + Lx2 + Lx + I(1, 2, •) ↰ α]α
 * I(1, 2, •)-Φ(1, •) ↰ α = [c + Lx2 + Lx2 + I(1, 3, •) ↰ α]α
 * I(1, •, 0)-Φ(1, •) ↰ α = [c + Lx2 + Lx·(I(2, 0, •) ↰ α)]α
 * I(2, 0, •)-Φ(1, •) ↰ α = [c + Lx22 + I(2, 1, •) ↰ α]α
 * I(2, 1, •)-Φ(1, •) ↰ α = [c + Lx22 + Lx + I(2, 2, •) ↰ α]α
 * I(2, •, 0)-Φ(1, •) ↰ α = [c + Lx22 + Lx·(I(3, 0, •) ↰ α)]α
 * I(•, 0, 0)-Φ(1, •) ↰ α = [c + Lx2·(I(1, 0, 0, •) ↰ α)]α
 * I(1, 0, 0, •)-Φ(1, •) ↰ α = [c + Lx3 + I(1, 0, 1, •) ↰ α]α
 * I(1, 0, •, 0)-Φ(1, •) ↰ α = [c + Lx3 + Lx·(I(1, 1, 0, •) ↰ α)]α
 * I(1, •, 0, 0)-Φ(1, •) ↰ α = [c + Lx3 + Lx2·(I(2, 0, 0, •) ↰ α)]α
 * I(•, 0, 0, 0)-Φ(1, •) ↰ α = [c + Lx3·(I(1, 0, 0, 0, •) ↰ α)]α
 * I(•, 0, 0, 0, 0)-Φ(1, •) ↰ α = [c + Lx4·(I(1, 0, 0, 0, 0, •) ↰ α)]α

M• (weakly Mahlos)
ΩL x + 1 = Ω• ↰ Lx = Ω• ↰ L• ↰ α = [c][c2]α


 * M• ↰ α = [c + ΩL x + 1 ]α
 * M(2, •) ↰ α = [c + ΩL x + 1 2]α
 * M(1, 0, •) ↰ α = [c + ΩL x + 1 2]α
 * M(1, 1, •) ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 ]α
 * M(2, 0, •) ↰ α = [c + ΩL x + 1 22]α
 * M(1, 0, 0, •) ↰ α = [c + ΩL x + 1 3]α

M•-I (M regular limits)

 * M•-I• ↰ α = [c + ΩL x + 1 + Lx]α
 * M•-I(2, •) ↰ α = [c + ΩL x + 1 + Lx2]α
 * M•-I(1, 0, •) ↰ α = [c + ΩL x + 1 + Lx2]α
 * M(2, •)-I• ↰ α = [c + ΩL x + 1 2 + Lx]α
 * M(2, •)-I(2, •) ↰ α = [c + ΩL x + 1 2 + Lx2]α
 * M(2, •)-I(1, 0, •) ↰ α = [c + ΩL x + 1 2 + Lx2]α
 * M(•, 0)-I• ↰ α = [c + ΩL x + 1 Lx]α
 * M(•, 0)-I(2, •) ↰ α = [c + ΩL x + 1 Lx2]α
 * M(•, 0)-I(1, 0, •) ↰ α = [c + ΩL x + 1 Lx2]α
 * M(1, 0, •)-I• ↰ α = [c + ΩL x + 1 2 + Lx]α
 * M(1, 0, •)-I(2, •) ↰ α = [c + ΩL x + 1 2 + Lx2]α
 * M(1, 0, •)-I(1, 0, •) ↰ α = [c + ΩL x + 1 2 + Lx2]α
 * M(1, 1, •)-I• ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 + Lx]α
 * M(1, 1, •)-I(2, •) ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 + Lx2]α
 * M(1, 1, •)-I(1, 0, •) ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 + Lx2]α
 * M(1, 1, •)-I(1, 1, •) ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 + Lx2 + Lx]α
 * M(1, •, 0)-I• ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 Lx]α
 * M(1, •, 0)-I(2, •) ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 Lx2]α
 * M(1, •, 0)-I(1, 0, •) ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 Lx2]α
 * M(1, •, 0)-I(1, 1, •) ↰ α = [c + ΩL x + 1 2 + ΩL x + 1 (Lx2 + Lx)]α
 * M(•, 0, 0)-I• ↰ α = [c + ΩL x + 1 2Lx]α
 * M(•, 0, 0)-I(2, •) ↰ α = [c + ΩL x + 1 2Lx2]α
 * M(•, 0, 0)-I(1, 0, •) ↰ α = [c + ΩL x + 1 2Lx2]α
 * M(•, 0, 0)-I(1, 1, •) ↰ α = [c + ΩL x + 1 2(Lx2 + Lx)]α

M•-Φ (M fixed points)

 * M•-Φ(1, •) ↰ α = [c + M•-I• ↰ α]α
 * M•-I•-Φ(1, •) ↰ α = [c + ΩL x + 1 + Lx + M•-I(2, •) ↰ α]α
 * M•-I(2, •)-Φ(1, •) ↰ α = [c + ΩL x + 1 + Lx2 + M•-I(2, •) ↰ α]α
 * M•-I(•, 0)-Φ(1, •) ↰ α = [c + ΩL x + 1 + Lx·(M•-I(1, 0, •) ↰ α)]α
 * M•-I(1, 0, •)-Φ(1, •) ↰ α = [c + ΩL x + 1 + Lx2 + M•-I(1, 1, •) ↰ α]α
 * M(•, 0, 0)-I(1, •, 0)-Φ(1, •) ↰ α = [c + ΩL x + 1 2(Lx2 + Lx·(M(•, 0, 0)-I(2, 0, 0) ↰ α))]α

Hyper-Mahlos

 * M(1; •) ↰ α = [c + ΩL x + 2 ]α
 * M(1; •)-I• ↰ α = [c + ΩL x + 2 + Lx]α
 * M(1; •)-I(2, •) ↰ α = [c + ΩL x + 2 + Lx2]α
 * M(1; •)-M• ↰ α = [c + ΩL x + 2 + ΩL x + 1 ]α
 * M(1; •)-M•-I• ↰ α = [c + ΩL x + 2 + ΩL x + 1 + Lx]α
 * M(1; •)-M•-I(2, •) ↰ α = [c + ΩL x + 2 + ΩL x + 1 + Lx2]α
 * M(1; •)-M(2, •) ↰ α = [c + ΩL x + 2 + ΩL x + 1 2]α
 * M(1; •)-M(2, •)-I• ↰ α = [c + ΩL x + 2 + ΩL x + 1 2 + Lx]α
 * M(1; •)-M(2, •)-I(2, •) ↰ α = [c + ΩL x + 2 + ΩL x + 1 2 + Lx2]α
 * M(1; 1, •) ↰ α = [c + ΩL x + 2 2]α
 * M(1; 2, •) ↰ α = [c + ΩL x + 2 3]α
 * M(1; 1, 0, •) ↰ α = [c + ΩL x + 2 2]α
 * M(2; •) ↰ α = [c + ΩL x + 3 ]α
 * M(3; •) ↰ α = [c + ΩL x + 4 ]α
 * M(4; •) ↰ α = [c + ΩL x + 5 ]α
 * M(•; 0)-I• ↰ α = [c + ΩL x2 ]α
 * M(•; 0)-Φ(1, •) ↰ α = [c + ΩL x + M(•; 0)-• ↰ α ]α
 * M(•; 0)-I(2, •) ↰ α = [c + ΩL x3 ]α
 * M(•; 0)-M• ↰ α = [c + ΩΩ L x + 1 ]α
 * M(•; 0)-M(2, •) ↰ α = [c + ΩΩ L x + 1 2 ]α
 * M(•; 0)-M(1, 0, •) ↰ α = [c + ΩΩ L x + 1 2 ]α
 * M(•; 0)-M(1; •) ↰ α = [c + ΩΩ L x + 2 ]α
 * M(•; 0)-M(•; 0)-I• ↰ α = [c + ΩΩ L x2 ]α
 * M(•; 0)-M(•; 0)-M(•; 0)-I• ↰ α = [c + ΩΩ Ω L x2  ]α

(from this place I strongly doubt that this is correct)
 * M(•; 0)-•-Φ(1, •) ↰ α = [c + Φ(1, Lx + 1)]α
 * M(1, 0; •) ↰ α = [c + IL x + 1 ]α
 * M(1, 0; 1, •) ↰ α = [c + IL x + 1 2]α
 * M(1, 0; 1, 0, •) ↰ α = [c + IL x + 1 2]α
 * M(1, 1; •) ↰ α = [c + ΩI L x + 1 + 1 ]α
 * M(1, 2; •) ↰ α = [c + ΩI L x + 1 + 2 ]α
 * M(1, •; 0)-I• ↰ α = [c + ΩI L x + 1 + Lx ]α
 * M(2, 0; •) ↰ α = [c + IL x + 2 ]α
 * M(2, •; 0)-I• ↰ α = [c + IL x2 ]α
 * M(1, 0, 0; •) ↰ α = [c + I(2, Lx + 1)]α
 * M(2, 0, 0; •) ↰ α = [c + I(3, Lx + 1)]α
 * M(1, 0, 0, 0; •) ↰ α = [c + I(1, 0, Lx + 1)]α

K

 * K• ↰ α = [c + ML x + 1 ]α
 * K•-I• ↰ α = [c + ML x + 1 + Lx]α
 * K•-M• ↰ α = [c + ML x + 1 + ΩL x + 1 ]α
 * K•-M(1; •) ↰ α = [c + ML x + 1 + ΩL x + 2 ]α
 * K•-M(1, 0; •) ↰ α = [c + ML x + 1 + IL x + 2 ]α
 * K(2, •) ↰ α = [c + ML x + 1 2]α
 * K(1, 0, •) ↰ α = [c + ML x + 1 2]α
 * K(1; •) ↰ α = [c + ΩM L x + 1 + 1 ]α
 * K(1, 0; •) ↰ α = [c + IM L x + 1 + 1 ]α
 * K(1| •) ↰ α = [c + ML x + 2 ]α
 * K(•| 0)-I• ↰ α = [c + ML x2 ]α
 * K(•| 0)-M• ↰ α = [c + MΩ L x + 1 ]α
 * K(•| 0)-K• ↰ α = [c + MM L x + 1 ]α
 * K(•| 0)-K(1; •) ↰ α = [c + MΩ M L x + 1 + 1 ]α
 * K(•| 0)-K(1, 0; •) ↰ α = [c + MI M L x + 1 + 1 ]α
 * K(•| 0)-K(1| •) ↰ α = [c + MM L x + 2 ]α
 * K(•| 0)-K(•| 0)-I• ↰ α = [c + MM L x2 ]α