Abstract
In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice (AC), we study partitions of Russell-sets into sets each with exactly n elements (called n -ary partitions), for some integer n. We show that if n is odd, then a Russell-set X has an n -ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell-set X such that |X | is not divisible by any finite cardinal n > 1 (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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