Coherent beliefs are not always types

For two interacting agents, we construct a space of nature states S and a coherent hierarchy of beliefs (σ-additive probability measures) of one agent about S, about S and the beliefs of the other agent about S, and so on - a hierarchy that has no σ-additive coherent extension over S and the hierarchies of the other agent. Thus, this hierarchy of beliefs cannot be the description of the beliefs of some type in some Harsanyi [Harsanyi, J.C., 1967-1968. Games with incomplete information played by Bayesian players, parts I, II, and III. Man. Sc. 14, 159-182, 320-334, 486-502] type space. Therefore, the space C of coherent hierarchies over S properly contains the universal space T* of 'all possible types' over S. We show how to extract T* out of C in a transfinite process.