The Limits to Gossip: Second-Order Shared Knowledge of All Secrets is Unsatisfiable

It is known that without synchronization via a global clock one cannot obtain common knowledge by communication. Moreover, it is folklore that without exchanging higher-level information arbitrary higher-level shared knowledge cannot be achieved. Here we make this result precise. We use epistemic logic to formally define “everyone knows that everyone knows that everyone knows all secrets” and then prove that this statement is unsatisfiable.