User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-5029411-20130809210243/@comment-1605058-20130810120704

Indescribable cardinals work a bit different - for every given type of formulas (like \(\Pi^2_1\)) if cardinal C is indescribable for this type then for every such formula P if we have P(C) then we have P(D) for some D<C. There is no simple analogue of this for up arrow notation.