The method of permutation models was introduced by Fraenkel in 1922 to prove the independence of the axiom of choice in set theory with atoms. We present a variant of the basic Fraenkel model in which supports are finite partitions of the set of atoms, rather than finite sets of atoms. Among our results are that, in this model, every well-ordered family of well-orderable sets has a choice function and that the union of such a family is well-orderable.
Bruce, Benjamin Baker
"A Permutation Model with Finite Partitions of the Set of Atoms as Supports,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 17:
1, Article 4.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss1/4