A note on dual Dedekind finiteness

TL;DR

This paper demonstrates, within ZF set theory, the existence of a family of sets where certain finite powers are dually Dedekind finite, but their immediate higher powers are dually Dedekind infinite, resolving an open question from 1974.

Contribution

It proves the consistent existence of a set family with alternating dual Dedekind finiteness and infiniteness in different powers, answering an open problem from prior research.

Findings

Existence of a family of sets with specific dual Dedekind finiteness properties.

Demonstrates consistency with ZF set theory.

Resolves an open question from 1974.

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 [J. Truss, Fund. Math. 84, 187--208 (1974)].