Worldly cardinal

In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ 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 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.[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 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 Vι.
  • The least κ with Vκ and Vι having the same theory.
  • The least κ with Lκ(Vκ) and Lι(Vι) having the same theory.
  • (not a worldly cardinal) The least κ with Vκ and Vι having the same Σ2 theory with real parameters.
  • (not a worldly cardinal) The least κ with Vκ ≺Σ2 Vι.
  • The least κ with Vκ ≺ Vι.
  • The least infinite κ with Vκ and Vι 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 Vι+1 does.
  • The least inaccessible cardinal ι.

References

  1. ^ Hamkins (2014).
  2. ^ Kanamori (2003), Theorem 1.3, p. 19.
  3. ^ Kanamori (2003), Lemma 6.1, p. 57.
  • 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


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