Some results on Specker's problem (Q1110506)

The following problem of Specker is investigated. Is it consistent that for every infinite set A, the power set P(A) is the union of countably many sets of size \(| A|\). It is proved that if it is consistent that there exists a regular limit of supercompacts then this is consistent for every set of successor cardinality. From a supercompact, it may be consistent that it holds for every limit cardinal plus for all but every third successor cardinal. The constructions use supercompact Prikry and Radin forcings with several observations extending earlier works by the authors.