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arXiv · 2412.07142

A note on dual Dedekind finiteness

Ruihuan Mao·Guozhen Shen
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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)].

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Ruihuan Mao, Guozhen Shen. 2024-12-10. A note on dual Dedekind finiteness. https://doi.org/10.1093/jigpal%2Fjzaf069

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