Worldly cardinal

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.^[1] A strong limit cardinal κ is worldly if and only if for every natural n, there is some ordinal θ < κ such that V_θ ≺_Σn V_κ.

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 (V_κ, V_κ+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]

The following are in strictly increasing order, where ι is the least inaccessible cardinal:

• The least worldly κ.
• The least worldly κ and λ (κ<λ, and same below) with V_κ and V_λ 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 V_κ ≺_Σ2 V_λ (this is higher than even a κ-fold iteration of the above item).
• The least worldly κ and λ with V_κ ≺ V_λ.
• 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 V_κ satisfying replacement for the language augmented with the (V_κ,∈) satisfaction relation.
• The least κ inaccessible in L_κ(V_κ); equivalently, the least κ>ω with V_κ satisfying replacement for formulas in V_κ in the infinitary logic L_∞,ω.
• The least κ with a transitive model M⊂V_κ+1 extending V_κ satisfying Morse–Kelley set theory.
• (not a worldly cardinal) The least κ with V_κ having the same Σ₂ theory as V_ι.
• The least κ with V_κ and V_ι having the same theory.
• The least κ with L_κ(V_κ) and L_ι(V_ι) having the same theory.
• (not a worldly cardinal) The least κ with V_κ and V_ι having the same Σ₂ theory with real parameters.
• (not a worldly cardinal) The least κ with V_κ ≺_Σ2 V_ι.
• The least κ with V_κ ≺ V_ι.
• The least infinite κ with V_κ and V_ι satisfying the same L_∞,ω statements that are in V_κ.
• The least κ with a transitive model M⊂V_κ+1 extending V_κ and satisfying the same sentences with parameters in V_κ as V_ι+1 does.
• The least inaccessible cardinal ι.