## Horst Herrlich, Paul Howard, Eleftherios Tachtsis

*On special partitions of Dedekind- and Russell-sets*

Comment.Math.Univ.Carolin. 53,1 (2012) 105-122.**Abstract:**A \emph{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 \emph{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.

**Keywords:** Axiom of Choice, Dedekind sets, Russell sets, generalizations of Russell sets, odd sized partitions, permutation models

**AMS Subject Classification:** 03E10 03E25 03E35 05A18

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