Does $\mathsf{DC}$ imply $\mathsf{AC}_ω$, uniformly?
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913635247325184 |
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| author | Andretta, Alessandro Notaro, Lorenzo |
| author_facet | Andretta, Alessandro Notaro, Lorenzo |
| contents | The Axiom of Dependent Choice $\mathsf{DC}$ and the Axiom of Countable Choice $\mathsf{AC}_ω$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\mathsf{DC}(X)$ asserts that any total binary relation on $X$ has an infinite chain, while $\mathsf{AC}_ω(X)$ asserts that any countable collection of nonempty subsets of $X$ has a choice function. It is well-known that $\mathsf{DC} \Rightarrow \mathsf{AC}_ω$. We study for which sets and under which hypotheses $\mathsf{DC}(X) \Rightarrow \mathsf{AC}_ω(X)$, and then we show it is consistent with $\mathsf{ZF}$ that there is a set $A \subseteq \mathbb{R}$ for which $\mathsf{DC} (A)$ holds, but $\mathsf{AC}_ω(A)$ fails. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06676 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Does $\mathsf{DC}$ imply $\mathsf{AC}_ω$, uniformly? Andretta, Alessandro Notaro, Lorenzo Logic 03E25 (Primary) 03E35, 03E40 (Secondary) The Axiom of Dependent Choice $\mathsf{DC}$ and the Axiom of Countable Choice $\mathsf{AC}_ω$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\mathsf{DC}(X)$ asserts that any total binary relation on $X$ has an infinite chain, while $\mathsf{AC}_ω(X)$ asserts that any countable collection of nonempty subsets of $X$ has a choice function. It is well-known that $\mathsf{DC} \Rightarrow \mathsf{AC}_ω$. We study for which sets and under which hypotheses $\mathsf{DC}(X) \Rightarrow \mathsf{AC}_ω(X)$, and then we show it is consistent with $\mathsf{ZF}$ that there is a set $A \subseteq \mathbb{R}$ for which $\mathsf{DC} (A)$ holds, but $\mathsf{AC}_ω(A)$ fails. |
| title | Does $\mathsf{DC}$ imply $\mathsf{AC}_ω$, uniformly? |
| topic | Logic 03E25 (Primary) 03E35, 03E40 (Secondary) |
| url | https://arxiv.org/abs/2305.06676 |