Worldly cardinal

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Worldly cardinal

A worldly cardinal κ is a cardinal such that the sets of rank less than κ, called V_κ, form a universe that satisfies the Zermelo–Fraenkel (ZF) axioms of set theory.

A handy way to recognize worldly cardinals is this: κ is worldly exactly when, for every natural number n, there are many ordinals θ < κ for which V_θ is a Σ_n-elementary submodel of V_κ. In plain terms, V_θ agrees with V_κ on all statements of a given logical complexity.

Zermelo’s categoricity theorem implies that every inaccessible cardinal is worldly.

Shepherdson’s theorem gives a stronger statement: being inaccessible is equivalent to the pair (V_κ, V_{κ+1}) being a model of second-order ZF set theory.

Worldly cardinals are not the same as inaccessible cardinals. In fact, the smallest worldly cardinal is singular (its cofinality is countable).

There is a strictly increasing sequence of worldly cardinals, with ι denoting the least inaccessible cardinal.