A full description of all expressible attitudes under common group knowledge modalities involves infinite hierarchies of higher-order information.

To fix all of the agents’ beliefs, the analysis moves to types \(\mathfrak{f}=\langle f_0,f_1,\ldots\rangle\), containing some \(f_n\) for every natural number n. In this extended framework the situation becomes more complicated. The space of all such types is universal in the following sense: every relational model can be mapped in a truth-preserving manner to the space of all types by sending each state to a full description of the agents’ corresponding first- and higher-order informational at