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]

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 (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]

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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