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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.05028 |
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| _version_ | 1866915373920550912 |
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| author | Pradic, Cécilia |
| author_facet | Pradic, Cécilia |
| contents | The Myhill isomorphism is a variant of the Cantor-Bernstein theorem. It states that, from two injections that reduces two subsets of $\mathbb{N}$ to each other, there exists a bijection $\mathbb{N} \to \mathbb{N}$ that preserves them. This theorem can be proven constructively. We investigate to which extent the theorem can be extended to other infinite sets other than $\mathbb{N}$. We show that, assuming Markov's principle, the theorem can be extended to the conatural numbers $\mathbb{N}_{\infty}$ provided that we only require that bicomplemented sets are preserved by the bijection. This restriction is essential. Otherwise, the picture is overall negative: among other things, it is impossible to extend that result to either $2 \times \mathbb{N}_{\infty}$, $\mathbb{N} + \mathbb{N}_{\infty}$, $\mathbb{N} \times \mathbb{N}_{\infty}$, $\mathbb{N}_{\infty}^2$, $2^{\mathbb{N}}$ or $\mathbb{N}^{\mathbb{N}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Myhill isomorphism theorem does not generalize much Pradic, Cécilia Logic Logic in Computer Science 03F50, 03F60 The Myhill isomorphism is a variant of the Cantor-Bernstein theorem. It states that, from two injections that reduces two subsets of $\mathbb{N}$ to each other, there exists a bijection $\mathbb{N} \to \mathbb{N}$ that preserves them. This theorem can be proven constructively. We investigate to which extent the theorem can be extended to other infinite sets other than $\mathbb{N}$. We show that, assuming Markov's principle, the theorem can be extended to the conatural numbers $\mathbb{N}_{\infty}$ provided that we only require that bicomplemented sets are preserved by the bijection. This restriction is essential. Otherwise, the picture is overall negative: among other things, it is impossible to extend that result to either $2 \times \mathbb{N}_{\infty}$, $\mathbb{N} + \mathbb{N}_{\infty}$, $\mathbb{N} \times \mathbb{N}_{\infty}$, $\mathbb{N}_{\infty}^2$, $2^{\mathbb{N}}$ or $\mathbb{N}^{\mathbb{N}}$. |
| title | The Myhill isomorphism theorem does not generalize much |
| topic | Logic Logic in Computer Science 03F50, 03F60 |
| url | https://arxiv.org/abs/2507.05028 |