Strong partition cardinal
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In Zermelo-Fraenkel set theory without the axiom of choice a strong partition cardinal is an uncountable well-ordered cardinal k such that every partition of the set [k]kof size k subsets of k into less than k pieces has a homogeneous set of size k.
The existence of strong partition cardinals contradicts the axiom of choice. The Axiom of determinacy implies that ℵ1 is a strong partition cardinal.
[edit] References
- Henle, J. M.; Kleinberg, E.M. & Watro, R.J. (1984), “On the Ultrafilters and Ultrapowers of Strong Partition Cardianls”, Journal of Symbolic Logic 49 (4): 1268-1272., <http://links.jstor.org/sici?sici=0022-4812%28198412%2949%3A4%3C1268%3AOTUAUO%3E2.0.CO%3B2-%23>
- Arthur W. Apter, James M. Henle, Stephen C. Jackson. "The Calculus of Partition Sequences, Changing Cofinalities, and a Question of Woodin." Transactions of the American Mathematical Society, vol. 352, no. 3 (1999), pp. 969-1003.
