Worldly cardinal
In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank "V" is a model of Zermelo–Fraenkel set theory. Relationship to inaccessible cardinals. By Zermelo's theorem on inaccessible cardinals, every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement that ("V", "V") is a model of second order Zermelo-Fraenkel set theory. Being worldly and being inaccessible are not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a singular cardinal.