View source for Strong partition cardinal

In [[Zermelo–Fraenkel set theory]] without the [[axiom of choice]], a '''strong partition cardinal''' is an [[uncountable]] [[well-ordered]] [[cardinality|cardinal]] <math>k</math> such that every [[partition (set theory)|partition]] of the set <math>[k]^k</math>of size <math>k</math> subsets of <math>k</math> into less than <math>k</math> pieces has a [[Homogeneous (large cardinal property)|homogeneous set]] of size <math>k</math>.

The existence of strong partition cardinals contradicts the axiom of choice. The [[axiom of determinacy]] implies that ℵ<sub>1</sub> is a strong partition cardinal.

==References==
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*{{citation|first1=James M.|last1= Henle|first2= Eugene M.|last2= Kleinberg|first3= Ronald J. |last3=Watro|title=On the ultrafilters and ultrapowers of strong partition cardinals|journal=[[Journal of Symbolic Logic]]|volume= 49|issue= 4 |year=1984|pages= 1268–1272|doi=10.2307/2274277|jstor=2274277|s2cid= 45989875}} 
*{{citation|first1=Arthur W.|last1=Apter|first2=James M.|last2=Henle|first3=Stephen C.|last3=Jackson|title=The calculus of partition sequences, changing cofinalities, and a question of Woodin|journal=[[Transactions of the American Mathematical Society]]|volume=352|issue=3|year=1999|pages=969–1003|jstor=118097|mr=1695015|doi=10.1090/S0002-9947-99-02554-4 |doi-access=free}}.
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[[Category:Cardinal numbers]]


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