This note proves the existence of dually Dedekind finite and infinite sets using ZF, suggesting new insights into cardinal classes.
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 Aₙ n∈ ω of sets such that, for all n∈ ω, Aₙⁿ is dually Dedekind finite whereas Aₙⁿ⁺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).
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Mao et al. (2025) studied this question.
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