User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-30004975-20180110054958/@comment-5529393-20180110085326

M(0) are the strongly critical ordinals, so Ξ(0,a) is just $$\Gamma_a$$.

M(1) are the regular cardinals, so Ξ(1,a) = $$\aleph_{a+1}$$, except when a is a weakly inaccessible cardinal, in which case Ξ(1,a) = a.

M(2) are the weakly Mahlo cardinals, so Ξ(2,0) is the smallest weakly Mahlo cardinal, Ξ(2,1) is the second smallest weakly Mahlo cardinal, and so on.

Ξ(3,a) is the ath smallest 2-weakly Mahlo cardinal. (Counting from 0 as usual)

Ξ(4,a) is the ath smallest 3-weakly Mahlo cardinal.

$$\Xi(\omega,\alpha\)$$ is the $$\alpha$$th smallest $$\omega$$-weakly Mahlo cardinal.

$$\Xi(K,\alpha)$$ is the $$\alpha$$th smallest cardinal $$\beta$$ that is $$\beta$$-weakly Mahlo.

and so on.