# Locally finite knowledge structures

With respect to the S5 multi-agent epistemic logic, we define a cell to be a minimal subset of knowledge structures known in common semantically by all the agents. A cell has finite fanout if at every knowledge strcuture every agent considers only a finite number of other knowledge structures to be possible. A set of formulas in common knowledge is finitely generated if the common knowledge of some finite subset implies the common knowledge of the whole set. For every finitely generated set of formulas held in common knowledge at some knowledge structure either this set determines uniquely a finite cell or there are uncountable many cells of finite fanout (and also uncountably many cells of uncountable size) at which exactly this set of formulas is known in common. The situation is very different, however, for sets of formulas held in common knolwedge that are not finitely generated - if there are uncountably many corresponding cells then either none of these cells or all of them could have finite fanout.