  It is difficult to say in a general way what makes a definition circular.  In this entry we are concerned exclusively with mathematical definitions of various sorts.  Consider the equation x = ½x + 1.  Is this a circular definition of the number 2?  In a sense, it is just that: a number has been defined in terms of itself.  But there is nothing problematic about this equation, and so one may wonder why this is in the same class of equations as x = x + 1, or x = x.  In the set theoretic setting, we often employ circular definitions and characterizations of sets and classes. For example, the collection HF of hereditarily finite sets may be characterized by 