Adam Brandenburger and H. Jerome Keisler (2006) introduce the following two person, Russel-style paradox. The statement of the paradox involves two concepts: beliefs and assumptions. An assumption is a player’s strongest belief: it is a set of states that implies all other beliefs at a given state. We will say more about the interpretation of an assumption below. Suppose there are two players, Ann and Bob, and consider the following description of beliefs.