In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory.Hamkins (2014). 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 (Vκ, Vκ+1) is a model of second order Zermelo-Fraenkel set theory.Kanamori (2003), Theorem 1.3, p. 19. Being worldly and being inaccessible are not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a singular cardinal.Kanamori (2003), Lemma 6.1, p. 57.
The following are in strictly increasing order, where is the least inaccessible cardinal: The least worldly κ. The least worldly κ and λ (κ<λ, and same below) with Vκ and Vλ satisfying the same theory. The least worldly κ that is a limit of worldly cardinals (equivalently, a limit of κ worldly cardinals). The least worldly κ and λ with Vκ ≺Σ2 Vλ (this is higher than even a κ-fold iteration of the above item). The least worldly κ and λ with Vκ ≺ Vλ. The least worldly κ of cofinality ω1 (corresponds to the extension of the above item to a chain of length ω1). The least worldly κ of cofinality ω2 (and so on). The least κ>ω with Vκ satisfying replacement for the language augmented with the (Vκ,∈) satisfaction relation. The least κ inaccessible in Lκ(Vκ); 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. (not a worldly cardinal) The least κ with Vκ having the same Σ2 theory as V. The least κ with Vκ and V having the same theory. The least κ with Lκ(Vκ) and L(V) having the same theory. (not a worldly cardinal) The least κ with Vκ and V having the same Σ2 theory with real parameters. (not a worldly cardinal) The least κ with Vκ ≺Σ2 V. The least κ with Vκ ≺ V. The least infinite κ with Vκ and V 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 V does. The least inaccessible cardinal .
References
External links Worldly cardinal in Cantor's attic