A note on dual Dedekind finiteness

Ruihuan Mao; Guozhen Shen

doi:10.1093/jigpal/jzaf069

paper

A note on dual Dedekind finiteness

arXiv:2412.07142 · doi:10.1093/jigpal/jzaf069

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 $ZF$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists a family $⟨ A_{n} ⟩_{n \in I ¨ }$ of sets such that, for all $n \in I ¨ $ , $A_{n}^{n}$ is dually Dedekind finite whereas $A_{n}^{n + 1}$ is dually Dedekind infinite. This resolves a question that was left open in [J. Truss, Fund. Math. 84, 187--208 (1974)].

6 pages