User blog:MilkyWay90/Help with understanding Veblen array notation


 * A. φ(0_0, 0_1, …, 1_α+1) = sup_≺{φ(0_0, 0_1, …, x_α) : x ∈ On} (Since I'm not using LaTeX, a_b means a with the b on the right-bottom corner next to it)
 * Three questions:
 * What is sup_≺ ? I know what sup is, but not with the _≺ 
 * What does 'the :' mean?
 * What does the set On mean?
 * B. φ(0_0,0_1,…,1_α)=sup_≺{φ(0_0, 0_1, …,1_β) : β < α} where α ∈ Lim
 * What does the set Lim mean?
 * C. φ((x+1)_0,y_1,…,z_β)≻φ(x_0,…,z_β), and there is no array between them
 * What is ≻?
 * The rest doesn't matter, as if people answer this, then I'll be able to understand the rest, so I didn't bother to format it.
 * D. φ(0_0,…,(x+1)_α+1,…,y_β)=sup_≺{φ(0_0,…,(x+1)_α,x_(α+1),…,y_β)}
 * E. φ(00,…,xα+1,yα+2,…,zβ)=sup≺{φ(00,…,wα+1,yα+2,…,zβ):w<x}φ(00,…,xα+1,yα+2,…,zβ)=sup≺{φ(00,…,wα+1,yα+2,…,zβ):w<x} where x∈Limx∈Lim
 * F. φ(00,…,(x+1)α,…,yβ)=sup≺{φ(00,…,1γ,…,xα,…,yβ):γ<α}φ(00,…,(x+1)α,…,yβ)=sup≺{φ(00,…,1γ,…,xα,…,yβ):γ<α} where α∈Limα∈Lim
 * G. φ(00,…,xα,…,yβ)=sup≺{φ(00,…,zα,…,yβ):z<x}φ(00,…,xα,…,yβ)=sup≺{φ(00,…,zα,…,yβ):z<x} where x,α∈Lim