Pcf and abelian groups (Q2855507)

The author proves a powerful combinatorial result in ZFC, which he calls the black box trichotomy theorem. The immediate application target is the establishment in ZFC of the trivial dual conjecture \(\text{TDU}_{\aleph_\omega}\) for abelian groups. For \(\mu\) an uncountable cardinal, \(\text{TDU}_\mu\) is the following statement: There is a \(\mu\)-free abelian group G, necessarily of cardinality \(\geq\mu\), such that Hom(G, Z) = {0}. For \(\mu=\aleph_n\), \(\text{TDU}_\mu\) holds in ZFC [\textit{S. Shelah}, Cubo 9, No. 2, 59--79 (2007; Zbl 1144.03034)].NEWLINENEWLINE To the author's disappointment, the status of \(\text{TDU}_{\aleph_\omega}\) is still open. Yet, he is able to ``show that it is very hard for V not to give a positive answer''; the failure of \(\text{TDU}_{\aleph_\omega}\) in ZFC places ``strong demands on cardinal arithmetic in many \(\beth_\delta\)''.NEWLINENEWLINE Here is a brief summary of the paper's contents: \S0 Background and basic definitions; \S1 Black box trichotomy theorem; \S2 Cases of weak GCH; \S3 Sufficient conditions for black box instances; \S4 \(\text{TDU}_\mu\) for R-modules.