User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-1605058-20180130220439

Agreed with Emlightened - higher-order quantification over the entire universe indeed is not very common, instead one only uses it for smaller structures.

A relevant example is that of a \(\Pi^n_m\)-indescribable cardinal. To save you the details, it roughly means that the cardinal \(\kappa\) in question is not uniquely described by any \(\Pi^n_m\) property. More precisely, if \(V_\kappa\) satisfies some \(\Pi^n_m\) formula, then \(V_\alpha\) does too for some \(\alpha<\kappa\).