Hierarchies of knowledge: An unbounded stairway

The rank of a partition space is the maximal ordinal number of steps in the process, in which events of the space are generated by successively applying knowledge operators, starting with events of nature. It is shown in Heifetz and Samet (1998) that this rank may be an arbitrarily large ordinal [Heifetz, A., Samet, D., 1998. Knowledge spaces with arbitrarily high rank. Games and Economic Behavior 22, 260-273]. Here we construct for each ordinal α a canonical partition space U_α, in analogy with the Mertens and Zamir (1985) hierarchical construction for probabilistic beliefs [Mertens, J.F., Zamir, S., 1985. Formulation of Bayesian analysis for games with incomplete information. Int. J. Game Theory 14, 1-29]. Our main result is that each partition space of rank α is embeddable as a subspace of U_λ, where λ is the first limit ordinal exceeding α.