implying from which, in the limit, we can infer common knowledge of . From this definition it can be seen that if is common knowledge, then is also common knowledge (). This syntactic characterization is given semantic content through so-called Kripke structures. A Kripke structure is given by a set of states (or possible worlds) S, n accessibility relations , defined on , intuitively representing what states agent i considers possible from any given state, and a valuation function assigning a truth value, in each state, to each primitive proposition in the language. The Kripke semantics for the knowledge operator is given by stipulating that is true at state s iff is true at all states t such that . The semantics for the common knowledge operator, then, is given by taking, for each group of agents G, the reflexive (modal axiom T) and transitive closure (modal axiom 4) of the , for all agents i in G, call such a relation , and stipulating that is true at state s iff is true at all states t such that . Set theoretic (semantic characterization) Alternatively (yet equivalently) common knowledge can be formalized using set theory (this was the path taken by the Nobel laureate Robert Aumann in his seminal 1976 paper). Starting with a set of states S. An event E can then be defined as a subset of the set of states S. For each agent i, define a partition on S, Pi. This partition represents the state of knowledge of an agent in a state. Intuitively, if two states s1 and s2 are elements of the same part of partition of an agent, it means that s1 and s2 are indistinguishable to that agent. In general, in state s, agent i knows that one of the states in Pi(s) obtains, but not which one. (Here Pi(s) denotes the unique element of Pi containing s. This model excludes cases in which agents know things that are not true.) A knowledge function K can now be defined in the following way: That is, Ki(e) is the set of states where the agent will know that event e obtains. It is a subset of e. Similar to the modal logic formulation above, an operator for the idea that "everyone knows can be defined as e". As with the modal operator, we will iterate the E function, and . Using this