Worldly cardinal
# 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.

## 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 $(V_κ,$ $V_{κ+1})$ 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 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 $ω_1$ (corresponds to the extension of the above item to a chain of length $ω_1).$
* The least worldly κ of cofinality $ω_2$ (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 $Σ_2$ 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 $Σ_2$ 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.
