In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory.[1]

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 (Vκ, Vκ+1) is a model of second order Zermelo-Fraenkel set theory.[2] Being worldly and being inaccessible are not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a singular cardinal.[3]


  1. ^ Hamkins (2014).
  2. ^ Kanamori (2003), Theorem 1.3, p. 19.
  3. ^ Kanamori (2003), Lemma 6.1, p. 57.