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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.16001 |
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| _version_ | 1866912917079719936 |
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| author | Khokhlov, Valentyn |
| author_facet | Khokhlov, Valentyn |
| contents | This paper introduces an alternative approach to proving the existence of choice functions for specific families of sets within Zermelo-Fraenkel set theory (ZF) without assuming any form on the Axiom of Choice (AC). Traditional methods of proving choice, when it is possible without AC, are based on explicit constructing a choice function, which relies on being able to identify canonical elements within the sets. Our approach, instead, employs the axiom schema of separation. We begin by considering families of well-ordered sets, then apply the schema of separation twice to build a set of possible candidates for the choice functions, and, finally, prove that this set is non-empty. This strategy enables proving the existence of choice function in situations where canonical elements cannot be identified explicitly. We then extend our method beyond families of well-ordered sets to families of sets, over which partial orders with a least element exist. After exploring possibilities for further generalization, we establish a necessary and sufficient condition: in ZF, without assuming AC, a choice function exists for a non-empty family if and only if each set admits a partial order with a least element. Finally, we demonstrate how this approach can be used to prove the existence of choice functions for families of contractible and path-connected topological spaces, including hyper-intervals in $\mathbb{R}^n$, hyper-balls, and hyper-spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16001 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Necessary and Sufficient Conditions for Proving Choice in Zermelo-Fraenkel Set Theory Khokhlov, Valentyn Logic 03E25 This paper introduces an alternative approach to proving the existence of choice functions for specific families of sets within Zermelo-Fraenkel set theory (ZF) without assuming any form on the Axiom of Choice (AC). Traditional methods of proving choice, when it is possible without AC, are based on explicit constructing a choice function, which relies on being able to identify canonical elements within the sets. Our approach, instead, employs the axiom schema of separation. We begin by considering families of well-ordered sets, then apply the schema of separation twice to build a set of possible candidates for the choice functions, and, finally, prove that this set is non-empty. This strategy enables proving the existence of choice function in situations where canonical elements cannot be identified explicitly. We then extend our method beyond families of well-ordered sets to families of sets, over which partial orders with a least element exist. After exploring possibilities for further generalization, we establish a necessary and sufficient condition: in ZF, without assuming AC, a choice function exists for a non-empty family if and only if each set admits a partial order with a least element. Finally, we demonstrate how this approach can be used to prove the existence of choice functions for families of contractible and path-connected topological spaces, including hyper-intervals in $\mathbb{R}^n$, hyper-balls, and hyper-spheres. |
| title | Necessary and Sufficient Conditions for Proving Choice in Zermelo-Fraenkel Set Theory |
| topic | Logic 03E25 |
| url | https://arxiv.org/abs/2602.16001 |