person knows that a second blue-eyed person knows that a third person has blue eyes, but no one knows that there is a third blue-eyed person with that knowledge, until the outsider makes their statement. In general: For k > 1, it is "(k − 1)th order" knowledge (). Each blue-eyed person knows that a second blue-eyed person knows that a third blue-eyed person knows that.... (repeat for a total of k − 1 levels) a kth person has blue eyes, but no one knows that there is a "kth" blue-eyed person with that knowledge, until the outsider makes his statement. The notion of common knowledge therefore has a palpable effect. Knowing that everyone knows does make a difference. When the outsider's public announcement (a fact already known to all, unless k=1 then the one person with blue eyes would not know until the announcement) becomes common knowledge, the blue-eyed people on this island eventually deduce their status, and leave. In particular: is free (i.e. known prior to the outsider's statement) iff . , with a passing day where no one leaves, implies the next day . for is thus reached iff it is reached for . The outsider gives for . Formalization Modal logic (syntactic characterization) Common knowledge can be given a logical definition in multi-modal logic systems in which the modal operators are interpreted epistemically. At the propositional level, such systems are extensions of propositional logic. The extension consists of the introduction of a group G of agents, and of n modal operators Ki (with i = 1, ..., n) with the intended meaning that "agent i knows." Thus Ki (where is a formula of the logical calculus) is read "agent i knows ." We can define an operator EG with the intended meaning of "everyone in group G knows" by defining it with the axiom By abbreviating the expression with and defining , common knowledge could then be defined with the axiom There is, however, a complication. The languages of epistemic logic are usually finitary, whereas the axiom above defines common knowledge as an infinite conjunction of formulas, hence not a well-formed formula of the language. To overcome this difficulty, a fixed-point definition of common knowledge can be given. Intuitively, common knowledge is thought of as the fixed point of the "equation" . Here, is the Aleph-naught. In this way, it is possible to find a formula