A note on dual Dedekind finiteness

Abstract A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. It is proved consistent with $\mathsf{ZF}$ (i.e. the Zermelo–Fraenkel set theory without the axiom of choice) that there exists a family $\langle A_{n}\rangle _{n\in \omega }$ of sets such that, for all $n\in \omega $, $A_{n}^{n}$ is dually Dedekind finite whereas $A_{n}^{n+1}$ is dually Dedekind infinite. This resolves a question that was left open in (Classes of Dedekind finite cardinals. Fund. Math. 1974;84:187–208).