In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank V_κ is a model of Zermelo–Fraenkel set theory. A strong limit cardinal κ is worldly if and only if for every natural n, there are unboundedly many ordinals θ < κ such that V_θ ≺_Σn V_κ.

By Zermelo's categoricity theorem, every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement that (V_κ, V_κ+1) 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.

The following are in strictly increasing order, where ${\displaystyle I}$ is the least inaccessible cardinal: