User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-11227630-20131105084532

Things like I, I(2), I(I), I(1,0), I(1,0,0,0) are inaccessible.

Things like M,M(2),M(M) are mahlo. They work as a diagonalizer of inaccessible "Veblen hierarchy" i.e. χ functions.

Things like M(1,0,0), M(1,0,0,0,0,0) are inaccessible mahlo.

Things like Ξ(3,0), Ξ(3,1), Ξ(3,Ξ(3,0)) are mahlo mahlo, or 1-mahlo. They work as a diagonalizer of inaccessible mahlo "Veblen hierarchy".

Things like Ξ(4,0), Ξ(4,1) are mahlo mahlo mahlo...

Things like K, K(2), K(K) are compact. They work as a diagonalizer of mahlo "Veblen hierarchy" i.e. Ξ functions.

Just imagine inaccessible compact ordinals, mahlo compact ordinals, incaccessible mahlo compact ordinals, mahlo mahlo compact ordinals, compact compact ordinals and compact compact compact ordinals...

And imagine this: inaccessible hierarchy is stage 1, mahlo hierarchy is stage 2, compact hierarchy is stage 3. Now think of stage α, and use a diagonalizer of "stage" functions!