The foregoing analysis of luck motivates the following analysis of epistemic luck: S is very lucky that her belief P is true in the actual world if P is false in at least one of the near close worlds. And S is lucky, but not as lucky, that her belief P is true if P is true in the actual world but false in at least one of the non-near close worlds. Stated otherwise, false belief in a very close world is incompatible with knowledge while false belief in a non-near close world is compatible with knowledge. Here is a formulation of the safety condition by Pritchard (2007, 2008, 2009a) as a non-circular necessary condition in a standard account of knowledge:
In set theory, singletons are "atoms" that have no (non-empty) proper parts; many consider set theory useless or incoherent (not "well-founded") if sets cannot be built up from unit sets. The calculus of individuals was thought to require that an object either have no proper parts, in which case it is an "atom", or be the mereological sum of atoms. Eberle (1970), however, showed how to construct a calculus of individuals lacking " atoms ", ., one where every object has a "proper part" (defined below) so that the universe is infinite.