In Zermelo–Fraenkel set theory without the axiom of choice a strong partition cardinal is an uncountable well-ordered cardinal such that every partition of the set of size subsets of into less than pieces has a homogeneous set of size .
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The existence of strong partition cardinals contradicts the axiom of choice. The Axiom of determinacy implies that ℵ1 is a strong partition cardinal.
References[edit]
- Henle, James M.; Kleinberg, Eugene M.; Watro, Ronald J. (1984), "On the ultrafilters and ultrapowers of strong partition cardinals", Journal of Symbolic Logic, 49 (4): 1268–1272, doi:10.2307/2274277, JSTOR 2274277, S2CID 45989875
- Apter, Arthur W.; Henle, James M.; Jackson, Stephen C. (1999), "The calculus of partition sequences, changing cofinalities, and a question of Woodin", Transactions of the American Mathematical Society, 352 (3): 969–1003, doi:10.1090/S0002-9947-99-02554-4, JSTOR 118097, MR 1695015.