The difference between common knowledge of formulas and sets

Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the k nowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many me mbers of the meet partition with this subset of formulas held in common kno wledge. If there are at least two agents, there are uncountably many member s of the meet partition where only the tautologies of the multi-agent S5 lo gic are held in common knowledge. Whether or not a member of the meet parti tion is the only one corresponding to a set of formulas held in common know ledge has radical implications for its topological and combinatorial struct ure.