On special partitions of Dedekind- and Russell-sets. (Q2898337)

The authors work in ZF and in ZFA (=Zermelo-Fraenkel (ZF) plus the existence of atoms) and the main stress is put on the absence of the axiom of choice (AC). All the results are surprising for everybody, whose set theory automatically includes choice. Let us give two samples. Recall: A Russell set (\(n\)-Russell set, resp.) is a set \(X\) admitting a partition \(\{X_i\: i\in \omega \}\) with each set \(X_i\) of size 2 (of size \(n\), resp.) such that no infinite subset of this partition has a choice function. The central problem, studied in the paper, is the question, whether an \(n\)-Russell set admits a \(p\)-ary partition, for \(p>1\), \(p\neq n\). The answer is complete for \(n=2\): If there is a Russell set and \(p>1\) is odd, then there is a Russell set which has a \(p\)-ary partition, and there is also a Russell set without a \(p\)-ary partition. Similar result holds for \(n=3\), while for bigger \(n\)'s the situation is different: There is a model of ZFA, in which for every \(p>6\), every 5-Russell set has a \(p\)-ary partition.NEWLINENEWLINEThe best summary of this paper was said in the referee's report: This paper is about one of the most bizarre areas of set theory.