Geodesic
On special partitions of Dedekind- and Russell-sets
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 105-122.

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A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal $a$ has a ternary partition (see Section 1, Definition 2) then the Russell cardinal $a+2$ fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell sets without ternary partitions. We then consider generalizations of this result.
Classification : 03E10, 03E25, 03E35, 05A18
Keywords: Axiom of Choice; Dedekind sets; Russell sets; generalizations of Russell sets; odd sized partitions; permutation models 
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Herrlich, Horst; Howard, Paul; Tachtsis, Eleftherios. On special partitions of Dedekind- and Russell-sets. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 105-122. http://geodesic.mathdoc.fr/item/CMUC_2012__53_1_a7/