2025/10/15 by Y. Hu, Hu, Yifan, Ruihuan Mao +3
Mathematics · #Advanced Topology and Set Theory #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2510.13508
A set A is dually Dedekind finite if every surjection from A onto A is injective; otherwise, A is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with ZF (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists an amorphous set A whose power set \mathscrP(A) is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss [J. Truss, Fund. Math. 84, 187--208 (1974)]. Nevertheless, we prove in ZF that, for all strictly amorphous sets A and all natural numbers n, \mathscrP(A)n is dually Dedekind finite, which generalizes a result of Goldstern.