Knowledge Spaces with Arbitrarily High Rank

The class of probabilistic belief spaces (Harsanyi, 1967-68,Man. Sci.,14,159-182, 320-324, 486-502) contains auniversal space, into which every other belief space can be mapped in a unique way by a belief morphism. We show that there is no analogous universal space in the class of knowledge spaces. To show this we define therankof a knowledge space, which is the ordinality of the most complicated descriptions of knowledge in the space. We then show that a knowledge space can be mapped by a knowledge morphism only to spaces of higher or equal rank. We construct knowledge spaces for arbitrarily high rank, demonstrating that there is no universal space.Journal of Economic LiteratureClassification Numbers: D80, D82.