Amorphous sets and dual Dedekind finiteness
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866917016485494784 |
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| author | Hu, Yifan Mao, Ruihuan Shen, Guozhen |
| author_facet | Hu, Yifan Mao, Ruihuan Shen, Guozhen |
| contents | A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists an amorphous set $A$ whose power set $\mathscr{P}(A)$ is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss [J. Truss, Fund. Math. 84, 187--208 (1974)]. Nevertheless, we prove in $\mathsf{ZF}$ that, for all strictly amorphous sets $A$ and all natural numbers $n$, $\mathscr{P}(A)^n$ is dually Dedekind finite, which generalizes a result of Goldstern. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Amorphous sets and dual Dedekind finiteness Hu, Yifan Mao, Ruihuan Shen, Guozhen Logic Primary 03E35, Secondary 03E10, 03E25 A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists an amorphous set $A$ whose power set $\mathscr{P}(A)$ is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss [J. Truss, Fund. Math. 84, 187--208 (1974)]. Nevertheless, we prove in $\mathsf{ZF}$ that, for all strictly amorphous sets $A$ and all natural numbers $n$, $\mathscr{P}(A)^n$ is dually Dedekind finite, which generalizes a result of Goldstern. |
| title | Amorphous sets and dual Dedekind finiteness |
| topic | Logic Primary 03E35, Secondary 03E10, 03E25 |
| url | https://arxiv.org/abs/2510.13508 |