Worldly cardinal


In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ 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 Vθ ≺Σn Vκ.

Relationship to inaccessible cardinals

By Zermelo's [categoricity theorem], every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement that 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 ι is the least inaccessible cardinal:
  • The least worldly κ.
  • The least worldly κ and λ with Vκ and Vλ satisfying the same theory.
  • The least worldly κ that is a limit of worldly cardinals.
  • The least worldly κ and λ with Vκ ≺Σ2 Vλ.
  • The least worldly κ and λ with Vκ ≺ Vλ.
  • The least worldly κ of cofinality ω1.
  • The least worldly κ of cofinality ω2.
  • The least κ>ω with Vκ satisfying replacement for the language augmented with the satisfaction relation.
  • The least κ inaccessible in Lκ; equivalently, the least κ>ω with Vκ satisfying replacement for formulas in Vκ in the infinitary logic L∞,ω.
  • The least κ with a transitive model M⊂Vκ+1 extending Vκ satisfying Morse–Kelley set theory.
  • The least κ with Vκ having the same Σ2 theory as Vι.
  • The least κ with Vκ and Vι having the same theory.
  • The least κ with Lκ and Lι having the same theory.
  • The least κ with Vκ and Vι having the same Σ2 theory with real parameters.
  • The least κ with Vκ ≺Σ2 Vι.
  • The least κ with Vκ ≺ Vι.
  • The least infinite κ with Vκ and Vι satisfying the same L∞,ω statements that are in Vκ.
  • The least κ with a transitive model M⊂Vκ+1 extending Vκ and satisfying the same sentences with parameters in Vκ as Vι+1 does.
  • The least inaccessible cardinal ι.