The axiom of choice for countable collections of countable sets does not imply the countable union theorem (Q1203787)

As has been shown by \textit{U. Felgner} [Comment. Math. Helv. 49, 114-124 (1974; Zbl 0283.02046)], in ZF the axiom of choice for families of nonempty well-orderable sets does not imply the countable union theorem. In Felgner's model \(\aleph_ 1\) is singular and the question arises if a similar independence result can be obtained by means of the Fraenkel- Mostowski method. In the paper under review the following theorem is proved by this method: The countable union theorem does not follow from the axiom of choice for well-ordered families of nonempty sets, each of cardinality not greater than or equal to \(2^{\aleph_ 0}\).