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