User blog comment:DontDrinkH20/Some Explanations of Certain Large Cardinals/@comment-11227630-20181019090840/@comment-11227630-20181021122909

By "rich structure" I mean there are many levels of the properties. Now I use indescribability as an example.

Using \(\Pi^n_m\)-indescribable cardinals we only have \(\omega^2\) kinds of indescribability. If for every level we can enumerate them (i.e. using "the \(\alpha\)-th \(\Pi^n_m\)-indescribable cardinal"), and put them into OCFs, the notational limit would be \(\psi(\psi_{I(\omega^2,0)}(0))\) (see the definition).

However, by "\(\alpha\) is \(\Pi^n_m\)-indescribable on \(A\)" where A is a class of ordinals (see the definition above in my "main comment"), there are much more cardinal properties. Firstly, ordinal \(\alpha\) is \(\Pi^0_1\)-indescribable on \(A\) iff \(\alpha=\sup(A\cap\alpha)\), i.e. \(\alpha\) is a limit of ordinals in A. An ordinal is \(\Pi^1_0\)-indescribable on \(A\) iff it is Mahlo on \(A\). Then the least instance of the following are ordered in increasing size:
 * 1) \(\Pi^1_0\)-indescribable (i.e. inaccessible cardinal)
 * 2) \(\Pi^0_1\)-indescribable on \(\Pi^1_0\)-indescribable cardinals (i.e. limit point of inaccessible cardinals) (though they don't work as a regular cardinal in OCF)
 * 3) \(\Pi^0_1\)-indescribable on "\(\Pi^0_1\)-indescribable on \(\Pi^1_0\)-indescribable cardinals" cardinals
 * 4) \(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_0\)-indescribable cardinals (i.e. 2-inaccessible cardinal)
 * 5) \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_0\)-indescribable cardinals" cardinals
 * 6) \(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_0\)-indescribable cardinals" cardinals (i.e. 3-inaccessible cardinal)
 * 7) \(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals (i.e. Mahlo cardinal)
 * 8) \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals" cardinals (i.e. limit point of Mahlo cardinals)
 * 9) \(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals" cardinals (i.e. inaccessible limit of Mahlo cardinals)
 * 10) \(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals, also \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals" cardinals (i.e. Mahlo limit of Mahlo cardinals)
 * 11) \(\Pi^1_0\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals" cardinals (i.e. 2-Mahlo cardinal)
 * 12) \(\Pi^1_0\)-indescribable on "\(\Pi^1_0\)-indescribable on '\(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals' cardinals" cardinals (i.e. 3-Mahlo cardinal)
 * 13) \(\Pi^1_1\)-indescribable (i.e. weakly compact cardinal)
 * 14) \(\Pi^0_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals (i.e. limit point of weakly compact cardinals)
 * 15) \(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals (i.e. inaccessible limit of weakly compact cardinals)
 * 16) \(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" (i.e. inaccessible limit of inaccessible limits of weakly compact cardinals)
 * 17) \(\Pi^1_0\)-indescribable on \(\Pi^1_0\)-indescribable cardinals, also \(\Pi^0_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals (i.e. Mahlo limit of weakly compact cardinals)
 * 18) \(\Pi^1_1\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals (i.e. weakly compact limit of weakly compact cardinals)
 * 19) \(\Pi^0_1\)-indescribable on "\(\Pi^1_1\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. limit of weakly compact limits of weakly compact cardinals)
 * 20) \(\Pi^1_1\)-indescribable, also \(\Pi^0_1\)-indescribable on "\(\Pi^1_1\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. weakly compact limit of weakly compact limits of weakly compact cardinals)
 * 21) \(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals (i.e. Mahlo on weakly compact cardinals)
 * 22) \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. limit of Mahlo on weakly compact cardinals)
 * 23) \(\Pi^1_1\)-indescribable, also \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. weakly compact limit of Mahlo on weakly compact cardinals)
 * 24) \(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable, also \(\Pi^0_1\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. "Mahlo on weakly compact" limit of Mahlo on weakly compact cardinals)
 * 25) \(\Pi^1_0\)-indescribable on "\(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. Mahlo on Mahlo on weakly compact cardinals)
 * 26) \(\Pi^1_1\)-indescribable, also \(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals (i.e. weakly compact cardinal that is Mahlo on weakly compact cardinals)
 * 27) \(\Pi^1_0\)-indescribable on "\(\Pi^1_1\)-indescribable, also \(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. Mahlo on "weakly compact cardinal that is Mahlo on weakly compact cardinals")
 * 28) \(\Pi^1_1\)-indescribable, also \(\Pi^1_0\)-indescribable on "\(\Pi^1_1\)-indescribable, also \(\Pi^1_0\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals (i.e. weakly compact cardinal that is Mahlo on "weakly compact cardinal that is Mahlo on weakly compact cardinals")
 * 29) \(\Pi^1_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals
 * 30) \(\Pi^1_0\)-indescribable on "\(\Pi^1_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals
 * 31) \(\Pi^1_1\)-indescribable, also \(\Pi^1_0\)-indescribable on "\(\Pi^1_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals
 * 32) \(\Pi^1_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals, also \(\Pi^1_0\)-indescribable on "\(\Pi^1_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals
 * 33) \(\Pi^1_1\)-indescribable on "\(\Pi^1_1\)-indescribable on \(\Pi^1_1\)-indescribable cardinals" cardinals
 * 34) \(\Pi^1_2\)-indescribable
 * 35) \(\Pi^0_1\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 36) \(\Pi^1_0\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 37) \(\Pi^1_1\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 38) \(\Pi^1_2\)-indescribable, also \(\Pi^0_1\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 39) \(\Pi^1_0\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 40) \(\Pi^1_1\)-indescribable, also \(\Pi^1_0\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 41) \(\Pi^1_2\)-indescribable, also \(\Pi^1_0\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 42) \(\Pi^1_1\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 43) \(\Pi^1_2\)-indescribable, also \(\Pi^1_1\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 44) \(\Pi^1_2\)-indescribable on \(\Pi^1_2\)-indescribable cardinals
 * 45) \(\Pi^1_3\)-indescribable
 * 46) \(\Pi^1_4\)-indescribable