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 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.

References

• Hamkins, Joel David (2014), "A multiverse perspective on the axiom of constructibility", Infinity and truth, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 25, Hackensack, NJ: World Sci. Publ., pp. 25–45, arXiv:1210.6541, Bibcode:2012arXiv1210.6541H, MR 3205072
• Kanamori, Akihiro (2003), The Higher Infinite, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag

External links

• Worldly cardinal in Cantor's attic