Strong partition cardinal

In Zermelo–Fraenkel set theory without the axiom of choice a strong partition cardinal is an uncountable well-ordered cardinal [math]\displaystyle{ k }[/math] such that every partition of the set [math]\displaystyle{ [k]^k }[/math]of size [math]\displaystyle{ k }[/math] subsets of [math]\displaystyle{ k }[/math] into less than [math]\displaystyle{ k }[/math] pieces has a homogeneous set of size [math]\displaystyle{ k }[/math]. The existence of strong partition cardinals contradicts the axiom of choice. The Axiom of determinacy implies that ℵ₁ is a strong partition cardinal.

References

• 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 
• 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 .