From Wikipedia, the free encyclopedia
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
.
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,
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.