 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 attitudes. This map, however, is usually not a modal bisimulation. In fact, the process of type construction could be continued indefinitely, yielding a transfinite hierarchy of mutually non-bisimilar type spaces (Heifetz &amp; Samet 1998). Such transfinite types can become relevant when the epistemic language is enriched with modalities for common group knowledge, in which case a full description of all expressible attitudes involves infinite hierarchies of higher-order information (Fagin, Geanakoplos et al. 1999). A recent logical study of type spaces including their probabilistic structure can be found in Bjorndahl and Halpern (2017).