Worldly cardinal

Short description: Large cardinal number

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 are unboundedly many ordinals 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 I 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 VI.
* The least κ with Vκ and VI having the same theory.
* The least κ with Lκ(Vκ) and LI(VI) having the same theory.
* (not a worldly cardinal) The least κ with Vκ and VI having the same Σ2 theory with real parameters.
* (not a worldly cardinal) The least κ with Vκ ≺Σ2 VI.
* The least κ with Vκ ≺ VI.
* The least infinite κ with Vκ and VI 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 VI+1 does.
* The least inaccessible cardinal I.

## References

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

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

## External links

* Worldly cardinal in Cantor's attic

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