Strong partition cardinal

Strong partition cardinal

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.


*citation|first=J. M.|last= Henle|first2= E.M.|last2= Kleinberg|first3= R.J. |last3=Watro|title=On the Ultrafilters and Ultrapowers of Strong Partition Cardianls|journal=Journal of Symbolic Logic|volume= 49|issue= 4 |year=1984|pages= 1268-1272.

* 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.

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